Post on 05-Jan-2017
Quelques problemes
de la theorie des systemes
paraboliques degeneres
non-lineaires
et des lois de conservation
Boris P. ANDREIANOV
These de doctorat en Mathematiques et Applications
soutenue
a l’Universite de Franche-Comte, Besancon, France
le 20 janvier 2000
devant le jury compose de
Assia Benabdallah, Universite de Franche-Comte
Philippe Benilan, Universite de Franche-Comte (le Directeur)
Valery Galkin, Institut de l’Energie Nucleaire, Obninsk, Russie (Rapporteur)
Raphaele Herbin, Universite de Provence, Marseille (Rapporteur)
Louis Jeanjean, Universite de Franche-Comte
Evgenii Radkevich, Universite Lomonossov de Moscou, Russie
Denis Serre, Ecole Normale Superieure de Lyon (Rapporteur)
Petra Wittbold, Universite Louis Pasteur de Strasbourg.
Remerciements
Je crois sincerement avoir eu une chance extraordinaire. Pendant les six dernieres
annees, Stanislav Nikolaevich Kruzhkov, puis Philippe Benilan, m’ont mene dans le monde
de la recherche. Parmi tout ce qu’ils m’ont donne, il m’est difficile de choisir le plus
precieux. Peut-etre, est-ce le gout pour la beaute des mathematiques; la volonte de ne pas
s’arreter a mi-chemin, d’aller au fond des choses. Et surtout, la confiance qu’ils m’ont
accordee, au-dela de ce que j’aurais pu esperer. A Stanislav Kruzhkov et Philippe Benilan,
j’adresse ma premiere pensee et ma plus profonde gratitude.
Je remercie beaucoup Evgenii Radkevich. Son soutien a aide a la poursuite de ce travail
en co-tutelle, les discussions avec lui ont ete stimulantes.
Athanasios Tzavaras s’est interesse a mes resultats; il m’a suggere un des problemes
abordes dans cette these. Je l’en remercie.
Denis Serre a accepte de donner son avis sur ce memoire; son appreciation m’est
tres importante. De plus, il m’a fait l’honneur de presider le jury de these. Je lui suis
doublement reconnaissant.
Je suis reconnaissant a Valerii Galkin, pour son rapport sur cette these, pour avoir
pris part a ce jury, et pour tout le soutien qu’il m’a accorde a Moscou.
Mes remerciements vont a Raphaele Herbin, pour l’avis qu’elle a donne sur ce travail,
pour ses explications, pour avoir accepte de participer au jury.
Je remercie Brian Gilding pour avoir accepte de faire un rapport sur ce memoire, et
pour l’attention qu’il a pretee a mes tous premiers resultats.
Assia Benabdallah m’a fait l’honneur de participer a ce jury. De longues discussions
avec elle, son encouragement constant m’ont ete precieux. Je lui adresse ma profonde
reconnaissance.
Je dois une partie importante de ma these a la collaboration avec Petra Wittbold. Elle
a bien voulu s’interesser aux diverses parties de ce memoire et prendre part au jury. J’en
suis honore. Je la remercie, et je remercie Michael Gutnic. Notre travail a trois m’a
beaucoup appris; il a ete un plaisir pour moi.
Louis Jeanjean a bien voulu s’interesser aux questions abordees dans cette these et
participer a ce jury. Ses remarques me seront tres utiles dans l’avenir; je le remercie de
son engagement.
La grande disponibilite de Catherine Pagani, Catherine Vuillemenot, Monique Digu,
Jacques Vernerey, Jean-Daniel Tissot, Odile Henri, Nathalie Pasquet m’a facilite la tache;
leur gentillesse m’a touche. Je leur dis “merci”.
Le Labo de Mathematiques de Besancon m’a accueilli pendant trois annees. Ou que
je sois, je serai toujours nostalgique de son ambiance. J’en garderai des souvenirs, j’en
garderai des amis.
Je dis “merci” et “spasibo” a mes amis : ceux de Moscou, de Besancon, de St-
Peterbourg... On a partage de bons moments; leur encouragement m’a beaucoup aide.
Ma mere et mon pere m’ont apporte un soutien inestimable. Je leur dedie cette these.
Resume
Dans la premiere partie, on traite par l’approche de viscosite auto-similaire le probleme
de Riemann pour la loi de conservation scalaire et les systemes de type “dynamique des
gaz isentropiques” en coordonnees de Lagrange et d’Euler. Dans chacun des cas, cette
etude aboutit aux resultats d’existence et d’unicite de la solution “wave-fan admissi-
ble” pour toute fonction de flux continue. En particulier, on couvre le cas d’apparition
du vide dans la dynamique des gaz et le cas des problemes mixtes avec transitions de
phase. D’autre part, pour une loi de conservation scalaire multi-dimensionnelle avec une
fonction de flux continue on demontre l’existence des solutions entropiques generalisees
maximum et minimum dans le cadre L1 ∩ L∞ . On etudie la nature de ces solutions a
l’aide de la theorie des semi-groupes non-lineaires; ensuite, on etend quelques resultats
d’unicite dus a Benilan et Kruzhkov.
Dans la deuxieme partie, on traite de systemes elliptiques-paraboliques dont les coeffi-
cients peuvent dependre de (t, x) . On demontre un theoreme de continuite des solutions
variationnelles par rapport aux donnees et obtient ainsi le resultat d’existence de Alt et
Luckhaus sous des hypotheses plus faibles, tout en mettant en evidence l’essentiel de
leurs arguments. On applique ensuite les techniques developpees pour demontrer la con-
vergence des schemas de volumes finis pour un systeme modele fortement nonlineaire,
qui apparaıt dans la physique des milieux poreux. On propose ainsi une approche pour
la convergence des methodes de volumes finis, ou la preuve se fait par reduction du cas
discret au cas continu.
Mots cles :
lois de conservation, probleme de Riemann, viscosite auto-similaire, dynamique des gaz
isentropiques, solutions entropiques generalisees, semi-groupes non-lineaires, systemes
elliptiques-paraboliques, conditions de type Leray-Lions, methodes de volumes finis.
Abstract
In the first part, one treats by the self-similar viscosity approach the Riemann problem
for a scalar conservation law and the systems of the “isentropic gas dynamics” type in
the Lagrange and Euler coordinates. In each of the cases, the study yields existence
and uniqueness of a wave-fan admissible solution for all continuous flux function. In
particular, the situation when vaccuum appears in gas dynamics is covered, as well as
the case of problems of mixed type with phase transitions. On the other hand, for a
scalar multidimensional conservation law with continuous flux function the existence of
maximum and minimum generalized entropy solutions in the L1 ∩ L∞ framework is
proved. Using the nonlinear semigroup theory, one studies the nature of these solutions;
then one extends some uniqueness results of Benilan and Kruzhkov.
In the second part, one treats elliptic-parabolic systems with coefficients that may
depend on (t, x) . A theorem on continuity of variational solutions with respect to data
is proved. This yields the existence result of Alt and Luckhaus under weaker hypotheses,
while clarifying the essence of their arguments. Necessary techniques are developped;
next, they are applied to proving convergence of finite volume schemes for a model
strongly nonlinear system, which appears in the study of porous media. An approach for
convergence of finite volume methods is proposed, where the proof goes on by reduction
of the discret case to the continuous one.
Key words :
conservation laws, Riemann problem, self-similar viscosity, isentropic gas dynamics, gen-
eralized entropy solutions, nonlinear semigroups, elliptic-parabolic systems, Leray-Lions
type conditions, finite volume methods.
Table des matieres :
Introduction
Chapitre I Enonces des “quelques problemes”
et un resume des resultats obtenus . . . . . . . . . . . . . . . . . . . . . . 13. . . 19
Chapitre II Les “quelques problemes”
dans le contexte mathematique et physique . . . . . . . . . . . . . 21. . . 31
Part 1. Conservation Laws with Continuous Flux Function
Chapter 1.I The Riemann Problem for Scalar Conservation Law
with Continuous Flux Function:
the Self-Similar Viscosity Approach . . . . . . . . . . . . . . . . . . . . . 35. . . 43
Chapter 1.II The Riemann Problem for p-System
with Continuous Flux Function . . . . . . . . . . . . . . . . . . . . . . . . . . 45. . . 62
Chapter 1.III On Viscous Limit Solutions to the Riemann Problem
for the Equations of Isentropic Gas Dynamics
in Eulerian Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63. . . 82
Chapter 1.IV L1 -Theory of Scalar Conservation Law
with Continuous Flux Function . . . . . . . . . . . . . . . . . . . . . . . . . . 83. . . 98
Part 2. Weak Solutions for Elliptic-Parabolic Systems
Chapter 2.I Elliptic-Parabolic problems: Existence and Continuity
with Respect to the Data of Weak Solutions . . . . . . . . . 101. . . 124
Chapter 2.II Convergence of Finite Volume Approximations
for a Nonlinear Elliptic-Parabolic Problem:
a Variational Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125. . . 159
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163. . . 172
Introduction
CHAPITRE I
Enonces des “quelques problemes”
et un resume des resultats obtenus†
Partie 1.Les lois de conservation
avec une fonction de flux continue
On etudie l’admissibilite et l’unicite de solutions pour la loi de conservation scalaire et les
systemes de type “elasticite non-lineaire” et “dynamique des gaz isentropiques”. En general,
la fonction de flux est seulement continue.
• Chapitre 1.I Le probleme de Riemann pour les lois de conservation scalaires avec une
fonction de flux continue : l’approche par viscosite auto-similaire.
· On traite le probleme suivant :
(CLε)
Ut + f(U)x = εtUxx,
U |t=0 =
u−, x < 0
u+, x > 0,
U : IR+ × IR 7→ IR, u± ∈ IR,
avec f(·) : IR 7→ IR continue.
· Les references principales sont :
Riemann [R1860], Rayleigh [Ray10], Hopf [H50], Gelfand [G59], Kalashnikov [Ka59], Dafer-
mos [D73a, D89], Kruzhkov [K69a, K69b, K70a], Kruzhkov, P.A.Andreyanov [KPA75],
Chapitre 1.IV.
†On presente dans ce Chapitre l’essentiel des resultats de la these. Les enonces et conditions exactes se
trouvent dans les chapitres correspondants dans les Parties 1 et 2.
14 Introduction
· Dans ce chapitre, on se donne pour objectif :
obtenir la solution admissible de (CL0) comme la limite des solutions de (CLε) quand
ε ↓ 0 , en donnant ainsi une preuve directe d’unicite pour (CL0) ; presenter le “wave fan
admissibility criterion” comme un critere bien adapte au cas de fonction de flux qui n’est
pas reguliere.
· Les principaux resultats sont :
- reduction de (CLε) avec ε > 0 a un probleme aux limites singulier, mais bien pose,
pour une equation differentielle ordinaire (Propositions 1,2);
- existence et unicite pour (CLε) avec ε > 0 (Theorem 1);
- convergence des solutions de (CLε) vers une limite, donnee par une formule explicite,
lorsque ε ↓ 0 , ce qui donne une demonstration directe d’unicite d’une solution “wave-fan
admissible” pour le probleme de Riemann (Theorem 2);
• Chapitre 1.II Le probleme de Riemann pour les p-systemes avec une fonction de flux
continue.
· On traite le probleme suivant :
(pSε)
Ut − Vx = 0
Vt − f(U)x = εtVxx,
(U, V )|t=0 =
(u−, v−), x < 0
(u+, v+), x > 0,
(U, V ) : IR+ × IR 7→ IR2,
u±, v± ∈ IR.
Dans un premier temps, f : IR 7→ IR est supposee continue strictement croissante
(Sections 1-3), puis la condition de monotonie est supprimee.
· Les references principales sont :
Dafermos [D74, D89], Leibovich [Le74], Krejcı,Straskraba [KrSt93], Tzavaras [Tz96],
James [Ja80], Shearer [Sh82], Slemrod [Sl89], Chapitre 1.I.
· Dans ce chapitre, on se donne pour objectif :
obtenir la solution admissible de (pS0) comme la limite des solutions de (pSε) lorsque
ε ↓ 0 ; inclure le cas hyperbolique-elliptique en toute generalite; expliciter le contexte
physique de l’admissibilite pour ce dernier probleme.
· Les principaux resultats sont :
- caracterisation de solutions de (pSε) , ε > 0 , et reduction a un probleme aux limites
pour une equation differentielle ordinaire (Lemma 1, Propositions 1,2,4 ; puis Lemma 4,
Propositions 5,6);
- unicite pour (pSε) , ε > 0 , et existence sous des hypotheses de croissance de f(·) en
±∞ (Theorem 1; puis Theorem 2);
0.I. Les enonces des problemes et un resume des resultats obtenus 15
- convergence des solutions de (pSε) vers une limite, donnee par une formule explicite,
lorsque ε ↓ 0 (Theorem 1; puis Theorem 2);
- observation que le “Riemann solver” obtenu permet de resoudre (pS0) d’une maniere
unique, mais ne correspond qu’a un nombre restreint de problemes physiques avec f(·)non-monotone (Remark 4).
• Chapitre 1.III Sur les limites visqueuses auto-similaires comme solutions du probleme
de Riemann pour les equations de la dynamique des gaz isentropiques en
coordonnees d’Euler.
· On traite le probleme suivant :
(GDε)
ρt + (ρu)x = 0
(ρu)t + (ρu2 + p(ρ))x = εtuxx,
(ρ, u)|t=0 =
(ρ−, u−), x < 0
(ρ+, u+), x > 0,
(ρ, u) : IR+ × IR 7→ IR+ × IR,
ρ± > 0, u± ∈ IR;
La fonction p : IR+ 7→ IR est supposee continue strictement croissante; en general, le
comportement de p(·) en zero n’interdit pas l’apparition du vide dans les solutions.
· Les references principales sont :
Dafermos [D89], Kim [Kim99], Slemrod, Tzavaras [SlTz89], Rozhdestvenskii, Janenko
[RoJa], Cheng, Hsiao [ChHs], Wagner [Wa87], Chapitres 1.I et 1.II.
· Dans ce chapitre, on se donne pour objectif :
construire les solutions de (GDε) avec ε > 0 qui peuvent contenir un point du vide, afin
de pouvoir resoudre (GD0) pour toutes donnees de Riemann; demontrer l’existence et
l’unicite pour (GDε) , ε > 0 ; obtenir la solution admissible de (GD0) comme la limite
des solutions de (GDε) lorsque ε ↓ 0 , et observer la formation du vide.
· Les principaux resultats sont :
- caracterisation des solutions de (GDε) pour ε > 0 , description des solutions avec
et sans le vide, et reduction de (GDε) , ε > 0 , a un probleme aux limites pour une
equation differentielle ordinaire (Lemma 1, Propositions 1,2,3, Lemmae 4,5,6);
- unicite pour (GDε) , ε > 0 et existence sous des hypotheses de croissance de p(·) en
+∞ (Theorem 1);
- convergence des solutions de (GDε) vers une limite, donnee par une formule explicite,
lorsque ε ↓ 0 (Theorem 2); observation de la structure de la solution admissible de
(GD0) , conditions necessaires et suffisantes d’existence d’une solution de (GDε) (for-
mule (38)) et d’apparition du vide (formule (39)).
16 Introduction
• Chapitre 1.IV La theorie L1 des lois de conservation scalaires avec une fonction de flux
continue†.
· On traite les problemes suivants :
(CP )
∂u
∂t+ divx φ(u) = g
u|t=0 = f,u : (0, T )× IRN 7→ IR,
(Eq) u+ divx φ(u) = f, u : IRN 7→ IR,
avec φ : IR 7→ IRN continue. Le cadre fonctionnel principal est c + L1 ∩ L∞ , c ∈ IR .
L’unicite d’une solution (c’est-a-dire, solution entropique generalisee) pour (CP ) et pour
(Eq) reste un probleme ouvert pour N ≥ 2 .
· Les references principales sont :
Benilan, Kruzhkov [BK96], Kruzhkov, Panov [KP90], Kruzhkov [K69a, K69b, K70a],
Benilan [B72], Crandall, Liggett [CL71], Benilan, Crandall, Pazy [BCP].
· Dans ce chapitre, on se donne pour objectif :
demontrer l’existence de la solution maximum et la solution minimum pour (CP ) et
(Eq) ; etablir le lien entre les solutions des deux problemes dans le cadre de la theorie
des semi-groupes non-lineaires; etablir des resultats partiels d’unicite, en particulier en
generalisant ceux de [BK96]; argumenter en faveur d’unicite en general.
· Les principaux resultats sont :
- l’existence d’une solution pour (Eq) et (CP ) dans le cadre L∞ (Lemma 2, Theo-
rem 6);
- l’existence de la solution maximum et la solution minimum pour (Eq) et (CP ) (The-
orem 1);
- la generation du semi-groupe “solution maximum de (CP)” par l’operateur “solution
maximum de (Eq) ”, accretif a domaine dense, et pareil pour les solutions minimum
(Proposition 2, Theorem 2, Remark 1);
- pour une fonction de flux φ(·) et une constante c donnees, l’equivalence de l’unicite
en general pour (Eq) et pour (CP ) (Corollary 1);
- pour une fonction de flux φ(·) donnee, l’unicite est garantie pour tout c ∈ IR , excepte
peut-etre un ensemble au plus denombrable (Propositions 1,3);
- l’unicite pour (Eq) et (CP ) dans le cas ou (N − 1) composantes de φ(·) sont
monotones, par recurrence sur la dimension d’espace N (Theorem 3);
- l’unicite pour (Eq) et (CP ) dans le cas ou les modules de continuite en zero des
composantes de φ(·+ c)− φ(c) satisfont la condition anisotropique de decroissance de
[KP90, BK96] (Theorem 4);
†Il s’agit d’un travail fait en commun avec Philippe Benilan et Stanislav N. Kruzhkov.
0.I. Les enonces des problemes et un resume des resultats obtenus 17
- malgre cela, et malgre le resultat de non-unicite dans le cadre L∞ de [KP90] qui
souligne la pertinence de la condition anisotropique dans L∞ , conclusion que la non-
unicite dans le cadre L1∩L∞ , le cas echeant, n’est pas intrinsequement liee au caractere
non-holderien de la fonction de flux (note en bas de page sous Theorem 4).
Partie 2.Solutions faibles pour
les systemes elliptiques-paraboliques
On etudie l’existence des solutions faibles et la convergence des solutions approchees pour
une classe de systemes paraboliques degeneres. La methode suit et eclaircit l’approche vari-
ationnelle suggeree par Alt et Luckhaus ([AL83]).
• Chapitre 2.I Problemes elliptiques-paraboliques : l’existence et la continuite par rapport
aux donnees de solutions faibles†.
· On traite les problemes du type suivant :
(Pr)
b(·, v)t = div a(·, v, Dv) + f(·, v) sur (0, T )× Ω ⊂ IR+ × IRd
b(·, v(·))|t=0 = u0 sur Ω
+ conditions aux limites mixtes
de Dirichlet h(·) et de Neumann g(·, v) sur (0, T )× ∂Ω.
Ici v : (0, T ) × Ω 7→ IRN , la fonction b : (0, T ) × Ω × IRN 7→ IRN est telle que
b(t, x, ·) est le gradient d’une fonction convexe differentiable Φ(t, x, ·) : IRN 7→ IR . On
prend des hypotheses appropriees sur les donnees b, a, f, g, h, u0 qui generalisent, pour
une partie, les conditions de [AL83] et [LJLL65] (les hypotheses (H1)−(H9) en general,
les hypotheses (1)−(3) dans le cas simple avec f, g, h qui s’annulent completement et
sans dependance en (t, x) ).
· Les references principales sont :
Alt, Luckhaus [ALpr, AL83], J.-L.Lions [JLL], Leray, J.-L.Lions [LJLL65], Kruzhkov [K69a],
Benilan, Wittbold [BW99].
· Dans ce chapitre, on se donne pour objectif :
obtenir un resultat d’existence et de dependance en donnees de solutions faibles pour (Pr)
avec les coefficients qui peuvent dependre de (t, x) ; mettre en evidence les arguments
essentiels dans la methode variationnelle proposee dans [ALpr, AL83]; discuter la necessite
des conditions de structure sur a , f , Φt .
†Il s’agit d’un travail fait en commun avec Philippe Benilan.
18 Introduction
· Les principaux resultats sont :
- une version du “chain rule” lemme, c’est-a-dire
(ChR)d
dt
∫
Ω
B(t, ·, v(t, ·)) =< b(t, ·, v(t, ·))t, v(t, ·) >E′,E −∫
Ω
Φt(t, ·, v(t, ·)),ou E est l’espace ou l’on cherche les solutions variationnelles, E ′ est son dual et
B(t, x, z) est
∫ z
0
(b(z)− b(ζ)) dζ (Lemma 1);
- une version de l’argument de compacite de [K69a], qui suggere que, pour une famille
de solutions d’equations d’evolution, la compacite dans L1 en (t, x) se deduit d’une
estimation L1 sur les translatees en x des solutions, a condition d’avoir des estimations
uniformes sur les solutions dans L1 d’une part, et sur la partie droite des equations dans
un espace de Sobolev negatif d’autre part (Lemma 6);
- une version appropriee de l’argument de Minty-Browder (Lemma 7);
- un theoreme de continuite pour les solutions faibles par rapport aux donnees de (Pr) ,
dans une topologie naturelle vis-a-vis des hypotheses prises, sous des hypotheses de
structure supplementaires (H11) − (H13) sur a , f et Φt et dans le cas h = 0 ;
dans la preuve de ce resultat, on applique successivement les trois arguments ci-dessus
et un argument qui combine l’equi-integrabilite avec le theoreme de Egorov (Theorem 1,
Remark 3);
- corollaires d’existence d’une solution faible pour (Pr) (Corollaires 1,2,3, Remark 4);
- indications sur la non-pertinence des restrictions imposes par les conditions de structure
(H11), (H12) pour l’existence d’une solution faible.
• Chapitre 2.II La convergence des approximations par les methodes des volumes finis
pour un probleme elliptique-parabolique non-lineaire : une approche
variationnelle†.
· On traite le probleme suivant :
(pL)
b(v)t = div ap(Dv) sur (0, T )× Ω ⊂ IR+ × IRd
b(v)|t=0 = u0 sur Ω
v = 0 sur ∂Ω,
pour un domaine polyhedral Ω ⊂ IRd , ou b est le gradient d’une fonction convexe
differentiable Φ : IRN 7→ IRN et div ap(D·) est un p-laplacien.
· Les references principales sont :
Alt, Luckhaus [ALpr, AL83], J.-L.Lions [JLL], Eymard, Gallouet, Herbin [EyGaHe], Chapitre
2.I
†Il s’agit d’un travail fait en commun avec Michael Gutnic et Petra Wittbold.
0.I. Les enonces des problemes et un resume des resultats obtenus 19
· Dans ce chapitre, on se donne pour objectif :
ecrire un schema de volumes finis pour (pL) , demontrer la consistance du schema et
proposer une approche de preuve de la convergence qui cherche a reduire le probleme au
cas continu du Chapitre 2.I autant que possible.
· Les principaux resultats sont :
- proposition d’une classe de schemas de volumes finis qui approche le gradient d’une
maniere satisfaisante vis-a-vis de la non-linearite de ap , puis d’un exemple d’un tel
schema (Definition 4, Remark 2);
- existence d’une solution discrete et estimations a priori (Theorem 2, Proposition 1);
- introduction des approximations “continues” de la solution discrete vh , du terme parabo-
lique b(vh) et du terme elliptique ap(“Dvh”) ayant de “bonnes proprietes”; reecriture
du systeme d’equations algebriques issu du schema des volumes finis sous une forme
d’equation dans D′ (Lemmae 2,5,6, Proposition 2);
- consistance de l’approximation par la methode des volumes finis de l’operateur div ap(D·)pour la classe de schemas proposee (Definition 5, Proposition 3, Theorem 3);
- convergence de vh , lorsque le pas de discretisation tend vers zero, vers une solution
faible de (pL) dans la classe de schemas proposee, qui est demontree par la methode
variationnelle du Chapitre 2.I appliquee aux approximations “continues” construites au-
paravant (Theorem 1).
CHAPITRE II
Les “quelques problemes” dans le contexte
mathematique et physique†
1. La theorie des equations aux derivees partielles, s’il y en a une, a pour points de depart “les
trois baleines”: les trois equations classiques. Ce sont l’equation hyperbolique (H) utt =
∆u , dite equation d’ondes; l’equation parabolique (P ) ut = ∆u , dite equation de la
chaleur; l’equation elliptique (E) ∆u = 0 , dite equation de Laplace. Il est habituel de
parler de caractere hyperbolique, parabolique ou elliptique dans des problemes plus generaux,
en se basant sur la ressemblance des methodes appliquees et le comportement des solutions.
Pour chacun des archetypes (H) , (P ) , (E) il existe un cadre specifique et une theorie
mathematique a part entiere, qui observe des phenomenes qui lui sont propres. Ainsi, (H)
et (P ) decrivent l’evolution en temps, tandis que (E) est relatif a la stabilite en espace.
Il y a un effet regularisant pour (P ) et (E) , c’est-a-dire, la regularite des solutions est
meilleure que celle des donnees; cela n’est pas le cas de (H) . Tandis que l’on observe
dans (P ) les phenomenes de propagation a vitesse infinie et de dissipation d’energie, (H)
presente les proprietes de conservation d’energie et de domaine de dependance fini. On
pourrait enumerer d’autres differences substantielles entre (H) , (P ) et (E) , et pourtant
ces equations lineaires sont les modeles les plus simples de phenomenes reels qui surgissent
en physique. Quand on veut concevoir des modeles plus generaux ou plus realistes, on
est souvent amene a considerer des problemes qui sont, d’une part non-lineaires, et d’autre
part, qui melangent en eux les comportements typiques pour (H) , (P ) , (E) . Le caractere
non-lineaire restreint dramatiquement les outils qui ont ete concus lors du traitement de
(H) , (P ) , (E) , et amene a en utiliser de nouveaux dans chaque cas qui se presente.
†On se donne pour objectif de presenter ici une vue generale sur l’ensemble des sujets abordes dans cette
these. Deux points de vue s’imposent : celui du contexte mathematique des equations aux derivees partielles
non-lineaires, mais aussi celui des motivations et des modeles physiques qui pesent considerablement sur le
sujet.
Apres un petit avant-propos, on fait le tour d’horizon des problemes traites. Il est suivi par une discussion
des motivations, methodes et resultats de chacune des deux parties de cette these. En conclusion, on explicite
un certain nombre des points d’interrogation que souleve le travail presente a Votre attention.
22 Introduction
Ainsi, j’ai ete confronte dans l’etude des problemes cernes dans le Chapitre 1 ci-dessus a
un nombre de difficultes qui proviennent de la non-linearite, de la non-regularite des donnees,
de changement ou d’imprecision sur le type des equations etudiees.
2. Les systemes hyperboliques des lois de conservation font l’objet de la premiere partie de
la these. La loi scalaire (CP ) , etudiee dans les chapitres 1.I et 1.IV, est tres interessante
en elle-meme. Elle correspond a un certain nombre de phenomenes, elle peut modeliser le
traffic routier par exemple; et elle represente le premier pas vers l’etude de systemes aussi
importants que ceux de la dynamique des gaz. Dans les chapitres 1.II et 1.III, on traite
le probleme qui a ete pose et resolu pour la premiere fois par B.Riemann ([R1860]) et qui
consiste a poursuivre la resolution du systeme de la dynamique des gaz isentropiques en
presence de discontinuites. Les systemes (pS0) , (GD0) concernes par les chapitres 1.II et
1.III modelisent les ecoulements isentropiques des gaz parfaits en coordonnees de Lagrange
et d’Euler, mais aussi l’elasticite non-lineaire et les fluides de van der Waals (cf. [R1860,
Ray10, CouFr, RoJa, Se, ChHs, Tz98, Ja80, Pe87]). Dans ce dernier cas en particulier, qu’on
traite en coordonnees de Lagrange, on est contraint a renoncer au caractere hyperbolique
du systeme, a cause de la presence des phases hyperboliques et elliptiques et des transitions
entre elles. Dans les coordonnees d’Euler, on considere aussi les cas d’apparition de l’etat du
du vide dans les solutions; le vide est la degenerescence specifique a certains systemes de la
dynamique des gaz. L’approche employee dans les chapitres 1.I-1.III necessite l’obtention de
resultats d’existence et d’unicite pour les problemes de Riemann (CLε) , (pSε) , (GDε) qui
sont les systemes regularises avec un terme parabolique lui-meme degenere.
De plus, on traite dans (CL0) , (pS0) , (GD0) et (CP ) le cas de fonction de flux
seulement continue. Ceci n’est en rien pathologique; en effet, une equation aussi simple
que (u3)t + (u2)x = 0 donne lieu a la loi de conservation scalaire wt + f(w)x = 0 avec
f : w 7→ w2/3 , qui n’est pas une fonction localement lipschitzienne. Le manque de regularite
de la fonction de flux entraıne des effets extremement interessants dans le cas d’une loi de
conservation scalaire multidimensionnelle. On observe le phenomene de propagation a vitesse
infinie, ce qui peut entraıner la perte du caractere conservatif de l’equation et la non-unicite
d’une solution dans la classe (pourtant tres bonne, cf. [K69a, K69b, K70a, PA71, KH74,
KPA75]) de solutions entropiques generalisees dans L∞ (cf. [B72, KP90, BK96]). La non-
unicite n’est peut-etre pas le cas dans le cadre de solutions L1 ∩ L∞ ; l’investigation de ce
dernier probleme, couplee avec une etude de l’equation (Eq) , et qui developpe les resultats
de [B72, KP90, BK96], fait l’objet du chapitre 1.IV.
D’un autre cote, l’unicite pour le probleme de Riemann dans le cas scalaire uni-dimension-
nelle (CL0) , pour une classe de solutions verifiant un critere d’admissibilite usuel depuis
[D74, D89] (et qui remonte aux idees de [Ray10, H50, G59, Ka59]), est etudiee directement
dans le chapitre 1.I. Des etudes tres similaires sont menes a bien dans chacun des chapitres 1.II
et 1.III. Ceci etend en particulier les resultats de [Ka59, Tz95, Sh82, Kim99].
0.II. Sur le contexte mathematique et physique 23
Des problemes lies aux degenerescences et singularites se presentent egalement dans les
equations des milieux poreux (cf. [Bear, AL83]). Les systemes que l’on considere peuvent
en particulier contenir des phases purement elliptiques. Par exemple, pour l’equation de
Richards, qui trouve d’importantes applications dans les sciences d’environnement, le terme
parabolique atteint le regime stationnaire en 1 (l’etat de saturation). Dans les regions
de degenerescence elliptique, l’evolution en temps d’une solution n’est que partiellement
controlee, ce qui ne permet pas d’obtenir de la compacite forte des solutions la ou c’etait le
cas pour les systemes paraboliques non-degeneres. Cela complique en particulier la question
de l’existence des solutions.
On fait une etude d’existence et de dependance continue pour les solutions faibles des
systemes elliptiques-paraboliques (Pr) dans le chapitre 2.I, en etendant en particulier le
resultat d’existence de [AL83]. On espere avoir atteint une plus grande clarte dans l’exposition
des arguments qui menent a l’existence d’une solution faible pour (Pr) , par rapport au papier
[AL83] qui est ici la reference cruciale.
Vu l’interet que suscite le developpement de schemas numeriques pour la modelisation
d’ecoulements dans les milieux poreux, on aborde dans le chapitre 2.II la convergence pour
les methodes des volumes finis, employees frequemment dans ce domaine (cf. [EyGaHe]).
On teste la convergence sur l’exemple d’une equation (en fait, d’un systeme) (pL) qui est
une forme simplifiee du modele de filtration d’un fluide en regime turbulent a travers un
milieu poreux, gouvernee par une loi de Darcy non-lineaire, ou encore d’ecoulement d’un gaz
turbulent dans un pipeline (cf. [DiDT94] et leurs references). A ma connaissance, il n’existe
pas d’autre resultat de convergence de schemas de volumes finis pour des equations de type
(pL) et, plus generalement, (Pr) dans le cas p 6= 2 ; en etudiant (pL) , on cherche a
debroussailler le chemin pour pouvoir aborder un jour des cas plus realistes.
3. Un probleme essentiel de la theorie des systemes hyperboliques des lois de conservation est
celui de l’unicite. C’est la theorie des solutions entropiques generalisees, dites g.e.s., qui resout
ce probleme pour les lois de conservation scalaires multi-dimensionnelles. La definition et les
methodes classiques, elaborees dans [K69a, K69b, K70a], donnent une theorie mathematique
tout a fait satisfaisante, au moins dans le cas de fonction de flux localement lipschitzienne.
On reviendra sur ce sujet dans la partie consacree au chapitre 1.IV. Avant, on aborde une
autre facette de la question de l’unicite pour les lois et systemes de conservation, qui est les
conditions d’admissibilite des chocs. Notons que la definition d’une g.e.s. se traduit par une
telle condition dans les solutions continues par morceaux (ou, plus generalement, dans les
solutions a variation localement bornee). Mais les conditions d’admissibilite des chocs sont
anterieures a la notion d’une g.e.s. (cf. [H50, O57, Lax57, G59]), et les deux sont reliees
par une motivation commune qui est la methode de viscosite. Ceci est important, vu les
recents developpements dans le domaine des chocs non-classiques (cf. [LF98]). On peut, en
effet, construire des theories non-classiques pour certaines lois de conservation non-convexes,
24 Introduction
et ou les solutions s’obtiennent comme limites d’approximations par la diffusion couplee a
une dispersion lorsque les deux tendent vers zero. Tout ce qui a ete dit exige une certaine
explication, ou plutot un detour historique. Je ne donne pas les references a beaucoup
de travaux importants qui ont amene a une bonne position du probleme mathematique de
resolution des lois de conservation; les noms des principaux intervenants et leurs contributions
sont contes dans l’introduction de l’article [D89].
La difficulte de la resolution globale en temps des systemes de la dynamique des gaz est
connue depuis le XIX siecle. Elle surgit a cause du caractere non-lineaire de propagation
d’ondes. A la difference de l’equation d’ondes (H), ou la regularite des donnees initiales
est preservee (faute d’effet regularisant), les solutions d’une loi de conservation non-lineaire
peuvent developper des discontinuites en temps fini meme lorsque les donnees initiales sont
analytiques. B.Riemann ([R1860]) semble etre le premier qui accepte des solutions discon-
tinues dans la dynamique des gaz isentropiques et construit une solution pour le probleme
d’evolution d’une discontinuite elementaire, qui porte son nom. En particulier, il ne choisit
pour la construction de la solution que “la moitie” des chocs envisageables, en pressentant
ainsi la condition d’admissibilite de Lax (cf. [Lax57]). Un bref resume du travail de Riemann
est donne dans [Se, Chap.4.6]. L’idee n’a pas fait l’unanimite. Rayleigh ([Ray10]) ecrit a pro-
pos de la condition de Rankine-Hugoniot, qui decrit les chocs envisageables dans une solution
discontinue: “however valid <the Rankine-Hugoniot condition> may be, its fulfillment
does not secure that the wave so defined is possible. As a matter of fact, a whole class
of such waves is certainly impossible, and I would maintain, further, that a wave of the
kind is never possible under the conditions, laid down by Hugoniot, of no viscosity or
heat conduction.†” C’est en retablissant une petite viscosite et/ou conductivite que Rayleigh
donne un sens a la distinction entre les “regimes permanents” (les chocs) admissibles et
inadmissibles; distinction qui n’a pas ete remarquee par Rankine lorsque ce dernier avait
deduit ses conditions de choc en partant des fluides conductifs. Le point de vue que defend
Rayleigh est, bien sur, celui du mecanicien. Il ne distingue pas le probleme mathematique du
phenomene physique que ce probleme est appele a decrire. La notion meme d’une solution
discontinue lui est etrangere: les “regimes permanents” ne sont pour lui rien d’autre que des
ondes de compression dans une region d’espace tres petite, et qui sont entretenues par la
dissipation d’energie dans la zone de transition.
Bien sur, aujourd’hui on peut donner tres facilement un sens purement mathematique a
un systeme de lois de conservation. C’est le sens faible, d’ailleurs plus naturel qu’une relation
differentielle dans le contexte physique de la conservation locale des grandeurs extensives.
On peut trouver beaucoup de solutions discontinues dans ce sens pour un meme probleme
†“aussi valide qu’elle < la condition de Rankine-Hugoniot> soit, rien ne garantit qu’une onde ainsi definie
soit possible dans le cas ou elle < la condition de Rankine-Hugoniot> est remplie. En fait, la classe entiere
de ces solutions est certainement impossible, et je maintiendrai, en outre, qu’une onde de la sorte n’est jamais
possible sous les conditions, imposees par Hugoniot, d’absence de viscosite et de transmission de la chaleur.”
0.II. Sur le contexte mathematique et physique 25
de Riemann, de sorte qu’il n’y a pas d’unicite. Ce qui distingue les differentes solutions
dans le cas de la dynamique des gaz, c’est leurs proprietes vis-a-vis de la conservation de
l’energie. Rayleigh semble etre le premier a observer que “maintenance of type in such a
<shock> wave <of condensation> requires removal of energy from the wave, while in
the contrary case of <shock> wave of rarefaction additional energy would need to be
supplied‡.” Ce qui permet de trancher, dans le cadre de modele considere, en accord avec
le deuxieme principe de thermodynamique dont l’importance avait ete negligee auparavant:
“although dissipative forces, such as those arising from viscosity, may possibly constitute
a machinery capable of maintaining the type of <shock> wave of condensation, in no
case they can maintain the type of <shock> wave of rarefaction§.” Du point de vue
moderne, on peut dire que Rayleigh postule implicitement que seules les limites des solutions
du systeme prenant en compte de petits effets dissipatifs sont admissibles comme solutions
faibles du systeme de la dynamique des gaz isentropiques. C’est de la que la theorie classique
des lois de conservation est partie quarante ans plus tard, avec le papier de Hopf [H50]
sur l’equation ut + (u2)x = 0 . Une solution faible du probleme de Cauchy pour cette
equation est construite comme la limite des solutions du meme probleme pour l’equation
ut + (u2)x = εuxx , que l’on peut resoudre explicitement. Il resulte du travail de Hopf que
seules les chocs qui joignent un etat u− a gauche a un etat inferieur u+ a droite sont
admissibles. Un moyen facile de “predire” ce dernier resultat est de considerer la possibilite
d’approximation de chocs par les ondes planes, approche qui fait d’ailleurs le titre de ce
meme papier de Rayleigh [Ray10]. Dans [G59], Gelfand montre deja comment resoudre le
probleme de Riemann pour une loi de conservation d’une maniere unique, en partant de cette
caracterisation par les ondes planes. Une autre variante de l’idee de viscosite apparait chez
Kalashnikov ([Ka59]). C’est cette version, vue a travers l’interpretation de Dafermos ([D89]),
qui donne lieu au chapitre 1.I, puis 1.II et 1.III ci-dessous.
C’est toujours l’idee de viscosite qui tend vers zero qui suggere la pertinence de la definition
d’une g.e.s. de Kruzhkov. Cette definition propose une caracterisation intrinseque d’une
solution; la motivation reste exterieure (Kruzhkov l’ecrit tres clairement dans ses notes de
cours [K70b], dont la diffusion reste malheureusement tres limitee). La theorie des g.e.s.
donne une reponse parfaite aux questions d’existence, d’unicite et de dependance continue
pour le cas d’une loi de conservation a fonction de flux localement lipschitzienne, dans la
classe de fonctions bornees, et cela en plusieurs dimensions d’espace. Helas, la generalisation
aux systemes de l’approche entropique sous forme de Lax (cf. [K70a, Lax71]) rencontre
‡“la maintenance d’une telle onde <de choc de condensation> necessite d’enlever de l’energie a l’onde,
tandis que dans le cas contraire d’une onde <de choc> de detente il faudrait lui livrer de l’energie addition-
nelle.”§“bien que les forces dissipatives, comme celles qui proviennent de la viscosite, pourraient constituer une
machinerie capable de maintenir le type d’onde <de choc> de condensation, en aucun cas elles ne peuvent
maintenir le type d’onde <de choc> de detente.”
26 Introduction
des difficultes importantes dans le cas des systemes 2× 2 , et incontournables au-dela. Ceci
incite a reprendre l’etude d’admissibilite des chocs. Or, on sait actuellement (cf. [Br96, Br99])
que la bonne condition d’admissibilite de chocs (cf. [Lax57]) pour la classe tres etudiee de
systemes vraiment nonlineaires ou lineairement degeneres (cf. [Lax57, Sm, Se]) peut en elle
seule donner l’unicite d’une solution. La question de l’unicite, tout comme la question de
l’existence (cf. [Gl65, Br99]), peut en fait etre ramenee a l’admissibilite pour les problemes
de Riemann. Pour les systemes strictement hyperboliques, il existe une construction (cf.
[Liu75, Liu81]) qui permet la resolution d’une maniere unique d’un probleme de Riemann
lorsque le saut dans les donnees est petit. La resolution globale peut etre beaucoup plus
compliquee dans chaque cas qui se presente.
Or, c’est une approche tres bien adaptee pour la resolution du probleme de Riemann que
propose Dafermos dans [D73a, D74, D89], et qui d’ailleurs permet de juger admissible ou non
une solution toute entiere, plutot que de la considerer choc par choc. L’unique solution d’un
probleme de Riemann ne peut dependre que du quotient x/t ; autrement dit, la solution est
auto-similaire. L’addition d’une viscosite comme celle dans (CLε) , (pSε) , (GDε) permet de
preserver cette propriete. L’idee de l’approche par viscosite auto-similaire est donc de ramener
la discussion d’un probleme de Riemann au niveau des equations differentielles ordinaires; aussi
une variete de methodes s’applique (cf. [D74, DDp76, Tz96], et [Tz98] pour un etat des lieux
dans ce domaine). Le programme habituel est de demontrer, par des methodes topologiques,
l’existence pour le probleme regularise; obtenir des estimations sur la variation des solutions,
puis passer a la limite par compacite; enfin, etudier la structure des limites vis-a-vis de la
possibilite d’approximation des ondes de choc par les ondes planes.
Tout en profitant de la forme auto-similaire, mais egalement de la structure elementaire
des systemes particuliers etudies, je propose dans les chapitres 1.I-1.III une approche differente
d’investigation des solutions. Les moyens mathematiques employes ici sont simples, pour
ne pas dire rudimentaires. Par une analyse de plus en plus laborieuse, on est amene dans
chacun des cas (CLε) , (pSε) , (GDε) a resoudre un probleme aux limites pour une equation
differentielle ordinaire. A chaque fois, cette equation admet un principe de maximum; celui-
ci me permet d’etablir des resultats tres precis d’existence, d’unicite et de convergence des
solutions lorsque ε ↓ 0 . Ainsi on demontre dans chaque cas que la viscosite auto-similaire
choisit une unique solution pour les problemes de Riemann. En plus, les formules explicites
(certes, connues) pour les solutions limites, que l’on obtient directement, permettent de
distinguer les chocs qui ne peuvent apparaıtre dans aucune solution et qu’on peut donc juger
comme inadmissibles; ceci remplace l’etude par les ondes planes.
Vu ce qui a ete dit sur le bien-fonde de la definition d’une g.e.s. et l’origine commune
des differentes branches de la theorie des systemes des lois de conservation, il est etonnant
qu’il existe une methode purement mathematique qui donne un sens intrinseque aux lois
de conservation. C’est d’ailleurs un des deux traits d’union entre les deux parties de cette
0.II. Sur le contexte mathematique et physique 27
these. Cette methode qui s’applique en meme temps aux lois de conservation scalaires et
aux equations elliptiques-paraboliques est l’approche par les semi-groupes non-lineaires. D’un
point de vue pratique, elle consiste a construire les solutions des equations d’evolution par le
procede de discretisation implicite en temps. Le trait commun des operateurs associes a une
loi de conservation scalaire et a une equation elliptique-parabolique coercive est la propriete
d’accretivite, qui donne la possibilite de continuer l’approximation a chaque pas de temps
d’une maniere stable. La theorie de semi-groupes non-lineaires peut permettre d’etablir
l’existence; mais en outre, elle peut donner l’unicite d’une solution dans les cas ou cette
unicite est un veritable probleme, (CP ) par exemple. Le procede meme de construction
des solutions semi-groupes (cf. [BCP]) inscrit la condition d’admissibilite dans les solutions;
l’irreversibilite en temps, suggeree dans les modeles physiques par la deuxieme loi de la
thermodynamique, est ainsi capturee (cf. [BCP88]).
Les resultats du chapitre 1.IV marquent davantage le lien qui existe entre les solutions
entropiques generalisees (cf. [K69a, K69b, K70a]) et les solutions semi-groupes (cf. [B72])
des lois de conservation. On utilise l’approche par semi-groupes non-lineaires pour etablir
la generation des semi-groupes des g.e.s. maximum, minimum de (CP ) par les operateurs
de la g.e.s. maximum, minimum de (Eq) , respectivement. On en deduit l’equivalence
entre l’unicite pour (CP ) et l’unicite pour (Eq) . Cela s’applique a la demonstration du
Theoreme 3 qui se fait par recurrence sur la dimension N d’espace et qui est le resultat
le plus interessant du chapitre 1.IV. Les autres techniques employees sont essentiellement
celles du papier [BK96]. L’existence des g.e.s. maximum et minimum, en absence presumee
d’unicite lorsque la fonction de flux n’est pas localement lipschitzienne, n’est pas un fait
trivial; on la prouve ici pour des donnees qui tendent vers une constante a l’infini. Comme
pour beaucoup d’autres resultats de ce travail, la demonstration est basee sur la possibilite
de prendre la fonction caracteristique de IRN tout entier comme la fonction test dans une
inegalite de type Kato (Lemma 3.1 dans [BK96]). Recemment, le resultat d’existence des
solutions maximum et minimum pour (CP ) et pour (Eq) a ete etendu au cas des donnees
bornees generales (cf. [P..]). Cela est d’autant plus interessant que l’on connaıt un exemple
de non-unicite (cf. [KP90]). Par contre, on remarque en passant que la non-unicite, le cas
echeant, d’une g.e.s. pour (CP ) et (Eq) dans le cadre L1 ∩L∞ doit etre gouvernee par
une propriete fine de la fonction de flux.
4. Jusqu’il y a quelques annees, l’unicite pour les systemes hyperboliques de lois de con-
servation manquait de resultats aussi importants que celui de Kruzhkov dans le cas scalaire.
C’est une analyse tres engagee de la structure d’ondes qui a permis a Bressan et ses col-
laborateurs ([Br96, Br99]) d’apporter une lumiere de comprehension a cette question. En
effet, j’ai deja mentionne les difficultes de l’approche entropique; l’approche par les semi-
groupes non-lineaires est freinee par le fait que l’on ne connaıt pas de norme pour laquelle
l’operateur associe au probleme d’evolution soit accretif. Cette meme difficulte surgit dans
28 Introduction
l’etude d’unicite pour les systemes elliptiques-paraboliques de type (Pr) , tandis que dans le
cas scalaire une reponse satisfaisante a la question d’unicite est donnee dans [Ot96, CaW99]
et [Ca99]. Dans ce cadre, la non-unicite, le cas echeant, est due a la degenerescence de la
partie parabolique et non a l’irregularite generique des solutions, comme c’est le cas pour les
systemes hyperboliques de lois de conservation.
Mais c’est encore l’existence d’une solution faible pour (Pr) qui souleve des questions.
La methode classique de construction d’une solution variationnelle (cf. [AL83]) exige les
hypotheses supplementaires (dites conditions de structure) de la dependance des coefficients
a, f en b(z) et non en z tout simplement. Or, l’approche par les semi-groupes non-lineaires
en particulier montre que ces conditions ne sont pas toujours indispensables (cf. [BW99]).
De plus, on trouve des indications a la meme conclusion tout en restant dans le cadre de
l’approche variationnelle de Alt-Luckhaus ([AL83]), qui est la methode du chapitre 2.I. Pas
tres importante du point de vue des applications physiques, la question de pertinence des
conditions de structure n’en reste pas moins obsedante. Je n’ai pas reussi a la resoudre,
bien qu’un travail important a ete fait dans le chapitre 2.I pour “minimiser” les restrictions
imposes sur (Pr) par l’approche de Alt-Luckhaus.
Cette methode prend sa premiere origine dans le travail [Bro63], qui se base sur la possi-
bilite d’appliquer l’approche de monotonie introduite dans [Mi62, Mi63] a la construction de
solutions aux equations elliptiques. Cette idee a ete etendue a une methode “de monotonie
et compacite” (cf. [JLL]) qui permet de resoudre par l’approche variationnelle, a travers la
methode de Galerkin, une classe de problemes elliptiques ([LJLL65]) et paraboliques ([JLL]).
Les conditions de Leray-Lions reapparaissent dans tout le procede de developpement de cette
approche variationnelle.
Dans le cas tres important de degenerescence elliptique d’un systeme parabolique, deux
autres arguments essentiels ont ete apportes dans [AL83]. Le premier (cf. aussi [Bam77])
impose a la fonction b dans (Pr) d’etre le gradient en z d’une fonction convexe; c’est le
“chain rule” lemme de type (ChR) . Il permet d’obtenir dans un premier temps les estima-
tions a priori qui remplacent celles de Leray-Lions, et de passer a la limite par l’argument de
Minty-Browder dans la phase finale de la demonstration. L’autre restriction que l’utilisation
de ce lemme impose se traduit dans les intrepretations physiques par la finitude de l’energie
dans l’etat initial. Le deuxieme argument est celui de la compacite L1 en temps et en espace
non pas pour les solutions v , mais pour les termes paraboliques b(v) . C’est ce deuxieme
argument qui est remplace dans le chapitre 2.I par un lemme assez ancien ([K69a]) et tres
bien adapte a la question qu’on se pose. L’utilisation de ce dernier lemme, qui est un resultat
de compacite tres general pour les equations d’evolution, sert aussi de trait d’union entre les
deux parties de cette these. C’est en effet ce lemme qui rend facile l’existence d’une g.e.s.
bornee pour le probleme de Cauchy pour une loi de conservation scalaire avec une fonction de
flux continue; une demonstration ecrite de ce resultat, qui semble manquer, est donnee dans
0.II. Sur le contexte mathematique et physique 29
les annexes du chapitre 1.IV. De la meme maniere, on donne dans les annexes du chapitre 2.I
les versions appropriees des trois arguments qui menent a l’existence pour (Pr) : le “chain
rule” lemme, le lemme de compacite dans L1 et l’argument de Minty-Browder.
Parallelement a la simplification de la preuve, on couvre dans le chapitre 2.I le cas de
dependance des coefficients dans (Pr) en (t, x) , en particulier de la dependance de b
en t (qui s’avere delicate, au point de donner lieu a un probleme qui est reste ouvert).
A ce moment un quatrieme argument apparaıt, une combinaison d’equi-integrabilite des
termes dans l’equation avec le theoreme de Egorov, et qui ramene la discussion au niveau
de coefficients uniformement continus en l’ensemble des variables. Quant aux conditions
de type Leray-Lions, on essaie d’imposer les hypotheses les moins restrictives dans le cadre
Lp(0, T ;W 1,p(Ω; IRN)) , choisi pour les solutions faibles. Une etude de (Pr) dans les espaces
d’Orlitz, a l’instar de [Kac90], doit permettre d’affaiblir certaines restrictions. Notons qu’on
ne traite pas dans le chapitre 2.I en toute generalite le cas ou la condition au bord sur
une des composantes de la solution est une condition de Neumann pure. D’une part, cela
exigerait une hypothese supplementaire sur la fonction b . D’autre part, bien que l’essentiel
de nos arguments s’applique aussi dans cette situation, on ne peut pas inclure ce cas dans le
theoreme principal sans nuire davantage a la lisibilite de la preuve.
Le resultat principal qu’on demontre dans le chapitre 2.I est que l’ensemble des solu-
tions faibles de (Pr) (faute d’en connaıtre l’unicite) est semi-continu inferieurement par
rapport aux perturbations des coefficients et des donnees de (Pr) dans la topologie na-
turelle. L’existence est obtenue comme un corollaire de ce resultat, en utilisant la methode
de Galerkin tout comme dans [JLL] and [ALpr]. C’est ce meme theoreme de continuite qui
donne la convergence des approximations de Galerkin.
Des problemes elliptiques-paraboliques modelisent, en particulier, des phenomenes qui
interessent l’industrie petroliere. D’ou l’importance des aspects numeriques. Les methodes
classiques utilises en resolution numerique de tels systemes, du type elements finis, ont un
inconvenient certain. C’est l’absence du caractere conservatif local. Par contre, les methodes
de type volumes finis sont concus pour assurer cette conservativite (cf. [EyGaHe]). A la
difference des methodes de Galerkin, on ne peut pas directement appliquer les resultats de
[AL83] ou du chapitre 2.I pour les volumes finis. Dans le chapitre 2.II, on s’est donne
pour objectif d’adapter le resultat du chapitre 2.I au cas d’approximation du systeme par les
methodes des volumes finis.
On a choisi le systeme (pL) , fortement non-lineaire, pour montrer la possibilite d’une
telle adaptation. Le chemin qui est souvent emprunte pour aborder un tel probleme est la
“discretisation” des arguments de la preuve du cas “continu”; en particulier, c’est le cas de
[EGH99], ou un resultat de convergence des approximations par un schema de volumes finis
est demontre pour l’equation de Richards, qui contient (pL) au cas scalaire pour p = 2 .
On a choisi l’approche inverse, c’est-a-dire, d’appliquer les arguments du cas continu aux
30 Introduction
versions “continues” de la solution discrete et du systeme d’equations algebriques qui la
determine. Le premier pas consiste donc a ecrire ce systeme sous la forme d’une equation
dans D′ . On arrive ensuite a garder la carcasse de la demonstration de la convergence, et
de ramener completement cette question de la convergence a la question de la consistance de
l’approximation de l’operateur elliptique par le schema choisi. La demonstration de la consis-
tance exige des nouveaux arguments, propres au cas discret, et impose des restrictions sur la
classe de schemas qu’on considere. Cela concerne tout particulierement le choix des moyens
d’approximation de la composante tangentielle du gradient de la solution aux interfaces des
volumes de controle. Ce probleme n’a pas de solution generalement admise; de plus, celles
que je connaıs ne sont pas suffisamment exactes pour nos besoins. On propose un exemple
d’approximation qui verifie nos hypotheses. Il faut pourtant indiquer que ces hypotheses, bien
qu’elles rendent la demonstration de la consistance plutot elegante, paraissent trop limitatives
du point de vue numerique.
5. A la fin, je citerai quelques-uns des problemes ouverts que laisse entrevoir ce memoire.
L’unicite d’une solution entropique generalisee pour une loi de conservation scalaire multidi-
mensionnelle avec une fonction de flux continue dans L1 ∩ L∞ demeure inconnue.
La pertinence des conditions de structure de type a(v,Dv) = a(b(v), Dv) pour l’existence
d’une solution faible au probleme (Pr) me paraıt tres douteuse. Cependant, je n’ai pas
reussi a demontrer qu’on peut s’en passer et avoir toutefois la convergence forte d’une suite
des solutions aux problemes approches, ne serait ce que dans le cadre de l’equation modele
b(v)t = vxx + F (v)x ([BW99]) ou l’existence est connue.
Le “chain rule” lemme (ChR) , qui est indispensable dans le chapitre 2.I, reste a demontrer
dans sa formulation naturelle. Cela permettrait d’avoir facilement un resultat plus general
dans le cas des conditions au bord de Dirichlet non-homogenes.
La possibilite de constuire une theorie consistante de solutions non-classiques de certaines
lois de conservation ([LF98]) souleve des questions interessantes, de par sa relation avec les
effets dissipatifs-dispersifs (que l’on peut sans doute prendre auto-similaires) d’une part, et
de par son interpretation eventuelle dans le cadre de la theorie des semi-groupes non-lineaires
d’autre part.
L’etude precise du probleme de Riemann pour des systemes speciaux, par les methodes de
viscosite auto-similaire, ne s’arrete peut-etre pas aux systemes de la dynamique des gaz.
Meme en presence des resultats generaux sur les systemes 2×2 ([DDp76]), on peut esperer
d’obtenir plus d’information, y compris des “Riemann solvers” plus ou moins explicites, dans
des cas tres particuliers. Le cas echeant, cela peut englober des systemes qui ne sont pas
hyperboliques ou qui donnent naissance aux ondes specifiques.
Le resultat de convergence des schemas de volumes finis qu’on propose me semble amu-
sant du point de vue purement mathematique; mais il est douteux qu’il soit accepte par la
0.II. Sur le contexte mathematique et physique 31
communaute des numericiens tant qu’il n’est pas etendu a des systemes plus realistes et a des
schemas moins contraignants. En particulier, il semble ([Ey..]) que le choix d’approximation
de la composante tangentielle du gradient sur les interfaces n’est pas tres important pour la
convergence des methodes de volumes finis. D’autre part, il serait interessant de voir ce que
pourrait apporter l’interpretation “continue” des schemas de volumes finis a l’etude d’autres
types de problemes.
Je continuerai a chercher des reponses a ces questions.
. . . no eto uge sovsem drugaja istorija.
En concluant cette introduction, je tiens a exprimer ma profonde reconnaissance
envers Stanislav Nikolaevıch Kruzhkov, qui m’a initie aux mathematiques, et je tiens
a remercier Philippe Benilan pour tout le soutien, d’ordre scientifique et d’ordre
humain, qu’il m’a apporte tout au long de notre connaissance, et qui m’a permis de
venir a bout de ce travail.
Part 1
Conservation Laws
with Continuous Flux Function
CHAPTER 1.I
The Riemann Problem
for Scalar Conservation Law
with Continuous Flux Function:
the Self-Similar Viscosity Approach†
Introduction
Consider the following Cauchy problem:
(1ε) Ut + f(U)x = εtUxx,
where f : IR → IR is continuous, U maps Π+ = IR+ × IR in IR , and ε ≥ 0 ;
(2) U(0, x) = U0(x) =
u−, x < 0
u+, x > 0
For the sake of simplicity, assume u− < u+ .
Our main concern is the Riemann problem for the scalar conservation law (10) . It is
well known that, because of the non-uniqueness of a weak solution to (10), (2) , additional
criteria have to be introduced in order to select the admissible one. We seek to establish the
uniqueness of a solution to (10), (2) satisfying the wave fan admissibility criterion, proposed
by C.M.Dafermos in [D89] in the context of general hyperbolic systems of conservation laws.
More exactly, we seek to distinguish the (wave-fan admissible) solution to (10), (2) as the
unique a.e. limit of solutions to (1ε), (2) as ε ↓ 0 . From this viewpoint, the term εtUxx
introduces vanishing artificial viscosity in (10), (2) , which we will call the self-similar viscosity.
A related approach to the admissibility for the Riemann problem has earlier been pursued
by A.S.Kalashnikov in [Ka59] (for the scalar case), V.A.Tupchiev in ([Tu64, Tu66, Tu73])
†This chapter is an extended English version of the note [BA2], which refines the approach used in [BA1].
36 Riemann problem for Scalar Conservation Laws
and C.M.Dafermos in [D73a, D74] (for the case of systems), and many others since then
(cf. the survey paper [Tz98]). The idea is to establish the existence and a uniform BV
bound for solutions Uε , ε > 0 ; pass to the limit by the compactness argument; derive
self-contained pointwise conditions on discontinuities of a limiting function, which is a weak
solution of (10), (2) ; and then infer the uniqueness when possible. In this way, the problem
under consideration has already been solved by A.S.Kalashnikov in [Ka59]. In the smooth
case, he had shown that any limit of solutions of (1εn), (2) as εn ↓ 0 fulfill conditions
that assure uniqueness for the Riemann problem (10), (2) (cf. (9) in Remark 3 at the end
of the chapter). As it is shown in [D73a], the most general form of such conditions is the
entropy-entropy flux admissibility (cf. [K70a, Lax71]).
For the case of scalar conservation law with continuous flux function, the wave-fan ad-
missibility for the Riemann problem is still equivalent to the entropy admissibility as defined
by S.N.Kruzhkov (see the notion of generalized entropy solution in Definition 1 below) in
[K69a, K69b, K70a] (cf. also A.I.Vol’pert, [V67]). This is due to the fact that there is unique-
ness of a generalized entropy solution to the Cauchy problem for (10) with general initial data
in L∞ even in the case the flux function is only continuous. Recall the following classical
definition.
Definition 1 A bounded measurable in Π+ function U(·, ·) is a generalized entropy solution
to the problem (10), (2) , if
(i) for all k ∈ IR , ψ ∈ C∞0 (Π+) such that ψ ≥ 0 one has
∫ ∫
Π+
|U(t, x)− k|ψt(t, x) + sign [U(t, x)− k][f(U(t, x))− f(k)]ψx(t, x) dxdt ≥ 0;
(ii) there exists a set E ⊂ IR+ of measure |E| = 0 such that for all t ∈ IR+ \ E the
function U(t, ·) is defined a.e. on IR , and for all r > 0 one has
limt∈R+\E, t→0
∫
|x|≤r
|U(x, t)− U0(x)|dx = 0.
While the general theory of the Cauchy problem for scalar conservation laws with Lipschitz
continuous flux function is due to S.N.Kruzhkov ([K69a, K69b, K70a]), results in the case of
only continuous flux function f(·) were first established by Ph.Benilan ([B72]) by the nonlinear
semigroup approach, which yields the existence and uniqueness for L1 initial data in the one-
dimensional case. The uniqueness of a generalized entropy solution for the case including the
Riemann problem for one-dimensional scalar conservation law with continuous flux function has
first been established by S.N.Kruzhkov and P.A.Andreyanov ([KPA75]). For general L∞ initial
data, the uniqueness in this case has been shown by L.Barthelemy ([Ba88]) and S.N.Kruzhkov,
E.Yu.Panov ([KP90]). Note that in the multidimensional case, the uniqueness is false in general
for L∞ data (cf. [KP90]) and still not clear for L1 ∩ L∞ data (cf. [BK96]). Some further
1.I.0. Introduction 37
results in this last direction are presented in Chapter 1.IV and [BABK]; while they have recently
been extended by E.Yu.Panov ([P99, P..]), the problem remains open.
It follows that for (10), (2) with only continuous f the approach of [Ka59] still could be
used. Arguing as in [K70a, K70b], one easily shows that a wave-fan admissible weak solution
of (10), (2) , whenever it exists, is a solution in the sense of Definition 1, so that it is the
unique generalized entropy solution. Note that, a fortiori, it is the unique solution to the
Riemann problem (10), (2) in the class of all wave-fan admissible solutions.
Here we bypass both the compactness and the entropy admissibility arguments, and prove
the uniqueness (and existence) of a wave-fan admissible weak solution independently, using
the somewhat elementary structure of the problem (10), (2) . The approach is essentially
one-dimensional (with respect to U as well as with respect to x ). Nevertheless, it can be
extended to special systems of conservation laws that have a kind of scalar structure (for the
system of isentropic gas dynamics, this is done in Chapters 1.II, 1.III herein) and where a
uniqueness result of general authority seems to be unknown.
The results of this chapter are summarized in Theorems 1,2 in Section 2. They state
that for all ε > 0 there exists a unique bounded self-similar solution Uε to (1ε), (2) ,
and Uε converge as viscosity vanishes. The limiting function U , which is the wave-fan
admissible solution to (10), (2) , is given by an explicit formula. Curiously, we do not need
the a priori knowledge of this formula for U . Indeed, the starting point for our approach is
the observation, going back at least as far as the lecture notes [G59] of I.M.Gelfand, on the
profile of distribution solutions to (10), (2) obtained through approximation of shock waves
by travelling waves. It can be summarized by saying that (in case u− < u+ ) the admissible
solution of (10), (2) is the graph inverse function to the derivative of the convex hull of f(·)on [u−, u+]
†, or else by the formula
(3) U(x/t) = ∂/∂x maxu−≤v≤u+
(xv − tf(v)).
These two assertions are actually equivalent, due to the Fenchel formula. A careful construction
of the “travelling waves”-admissible solution in case of smooth f(·) can be found in lecture
notes [K70b] by S.N.Kruzhkov (see also [GR], and [ChHs] and references therein); in [GiNTe],
the profile of the solution is directly obtained from Definition 1. Writing U under the form
chosen above permits to compare the formula (3) with formulae, suggested by the convex
analysis, that were proposed in [H50, Lax57, KPe87] for different f(·) and U0(·) . Due to
the uniqueness of a generalized entropy solution, they are all equivalent in the case of Riemann
problem with regular convex flux function.
In the lengthy Remark 3 at the end of the chapter some comments on interrelations of
the entropy admissibility, usual vanishing viscosity, self-similar viscosity and approximation by
†We refer to the greatest convex function F (·) on [u−, u+] such that F ≤ f as to the convex hull of
f(·) on [u−, u+] .
38 Riemann problem for Scalar Conservation Laws
travelling waves are presented.
1 Restatement of the problem
First let ε > 0 be fixed. Let restrict our attention to bounded self-similar solutions of
(1ε), (2) . For simplicity, we assume in the sequel that u− < u+ . Integrating the differential
equation for U(ξ) = U(t, x) , where ξ = x/t , we arrive to the following definition.
Definition 2 Let ε > 0 . A solution of (1ε), (2) is a function U(t, x) = U(x/t) , U ∈C1(IR) , that verifies
(4) εU ′(ξ) = −∫ ξ
ξ0
ζU ′(ζ)dζ + f(U(ξ)) +K with some ξ0, K ∈ IR, and U(±∞) = u±.
The strict monotony property for solutions of (4) , trivial in the case of smooth flux
function, can fail if f(·) is only continuous (cf. Remark 2). Nevertheless, we have the
following result.
Lemma 1 All solution U(ξ) of the problem (4) is non-decreasing on IR .
Proof: Assume the contrary. Then there exists c a point of extremum of U(·) on IR such
that U(c) = u0 6= u± . For definiteness, assume that c is a point of local maximum. Consider
the greatest segment [c1, c2] containing c such that U |[c1,c2] ≡ u0 . For all α > 0 small
enough there exist ξ1 = ξ1(α) , ξ2 = ξ2(α) such that ξ1 < c1 ≤ c2 < ξ2 , U(ξ1) = U(ξ2) =
u0 − α , and the distance between ξ2 and ξ1 is the least possible. Since U ∈ C1(IR) ,
there exists maxξ∈[ξ1,ξ2] |U ′(ξ)| = |U ′(ξ)| =M > 0 with some ξ ∈ [ξ1, ξ2] . For definiteness,
assume that c2 < ξ ≤ ξ2 Consider separately the two possibilities.
a) U ′(ξ) = −M . In this case U(ξ1) = U(ξ2) ≤ U(ξ) < u0 = U(c1) . Take ξ2 = ξ
and ξ1 = maxξ| ξ1 ≤ ξ < c1,U(ξ) = U(ξ) . One has U > U(ξ1) on [ξ1, c1] , therefore
U ′(ξ1) = m ≥ 0 . Taking into account that U ′|[c1,c2] ≡ 0 , and using (4) with ξ0 = c and
the corresponding constant K = Kc , one obtains
εU ′(ξ1) = −∫ ξ1
c1
ζU ′(ζ)dζ + f(U(ξ1)) +Kc,
εU ′(ξ2) = −∫ ξ2
c2
ζU ′(ζ)dζ + f(U(ξ2)) +Kc.
Subtracting these two equalities, one finds that ε(M +m) ≤ M · (|ξ1| + |ξ2|) · ((c1 − ξ1) +
(ξ2 − c2)) .
b) U ′(ξ) =M > 0 . According to the choice of ξ2 , one has ξ < ξ2 and U(ξ) > U(ξ2) , so
that there exists ξ2 = minξ| ξ < ξ < ξ2,U(ξ) = U(ξ) . It follows that U ′(ξ2) = −m ≤ 0 .
Set ξ1 = ξ and argue as in case a).
1.I.1. Restatment of the problem 39
In the two cases, we infer that ε ≤ (|ξ1|+ |ξ2|) · ((c1 − ξ1) + (ξ2 − c2)) . As α → +0 , one
has ξ1 → c1 , ξ2 → c2 , so that ε ≤ 0 at the limit. This contradiction proves the lemma.
⋄
By Lemma 1, the function Ξ(u) = [U(ξ)]−1 is defined a.e. on [u−, u+] and monotone.
Proposition 1 Assume that a function U is a solution of (1ε), (2) in the sense (4) . Then
the function Φ(u) =
∫ u
U(0)
Ξ(v)dv −K on [u−, u+] is a solution of the following problem :
(5)
Φ ∈ C[u−, u+] and Φ is convex
Φ(u) ≤ f(u) on [u−, u+] and Φ(u±) = f(u±)ε
f−Φ∈ L
loc1 (u−, u+)
Φ(u) ≥ εf(u)−Φ(u)
(f(u)− Φ(u))(Φ(u)− ε
f(u)−Φ(u)
)= 0
in the sense of measures
on (u−, u+)
(in the rest of the chapter, ˙ stands for d/du ).
Conversely, assume Φ(·) is a solution of (5) . Then the function
U(t, x) = U(x/t) = [Φ(u)]−1 ≡ ∂/∂x maxu−≤v≤u+
(xv − tΦ(v))
is a solution of (1ε), (2) in the sense (4) .
Proof: (4) ⇒ (5) Rewrite the equation (4) under the form
(6) εU ′(ξ) = f(U(ξ))− Φ(U(ξ));
it follows that Φ ∈ C(u−, u+) and Φ ≤ f . Since Φ(u) = Ξ(u) is an a.e. continuous,
non-decreasing function, Φ is convex. Set Ω = u| ∃ξ : U(ξ) = u, U ′(ξ) = 0 . The
Lebesgue measure |Ω| is zero, by the Sard lemma, and Ω ≡ u|Φ(u) = f(u) . For all
u ∈ [u−, u+] \ Ω there exists Φ(u) = Ξ(u) = 1/U ′(ξ) > 0 ; we have Φ(u) = εf(u)−Φ(u)
and Φ(u) < f(u) . Since (f − Φ)Φ = 0 on Ω , it follows that (f − Φ)(Φ− ε
f−Φ
)= 0
in the sense of measures on (u−, u+) . Since |Ω| = 0 and Φ ≥ 0 , one has Φ ≥ εf−Φ
on (u−, u+) in the same sense. Consequently, for all segment [a, b] ⊂ (u−, u+) one has∫ b
aε
f(u)−Φ(u)du ≤ Φ(b + 0) − Φ(a − 0) < ∞ , so that ε
f−Φ∈ L
loc1 (u−, u+) . The equation
(6) , together with U(±∞) = u± , imply that the limits limξ→±∞
U ′(ξ) exist and are zero, so
that Φ(u±) = f(u±) and Φ ∈ C[u−, u+] .
(5) ⇒ (4) Define the multivalued function Ξ(·) by Ξ : u ∈ [u−, u+] 7→ [Φ(u−0), Φ(u+
0)] , with Φ(u± ± 0) = ±∞ . Set Ω := u|Φ(u) = f(u) . It is clear that Ξ is strictly
increasing and single-valued on the complementary of Ω . Let U = [Ξ]−1 ; one has U ∈C(IR) . If u0 /∈ Ω , there exists Ξ(u) = ε
f(u)−Φ(u)> 0 in a neighbourhood of u0 , so that
(6) is satisfied at the point ξ0 = Ξ(u0) . If u0 ∈ Ω and ξ0 ∈ Ξ(u0) , then for all α > 0
40 Riemann problem for Scalar Conservation Laws
there exists a neighbourhood of u0 such that f(u)− Φ(u) < εα ; consequently, Φ > 1/α
in this neighbourhood. It follows that |Ξ(u0 + δ) − ξ0| ≥ |δ|/α for all δ small enough;
therefore there exists U ′(ξ0) = 0 and (6) is satisfied in all the cases. Thus U ∈ C1(IR)
and (6) implies (4) , by the definition of U(·) . Clearly, U(±∞) = u± . By the Fenchel
formula, it follows that the other representation of U holds. ⋄
Remark 1 It is interesting to observe that , according to Proposition 1, the function w(t, x) =
Φ(U(t, x)) satisfies the Hopf equation ([H50]) wt +wwx = 0 if U is a solution of (1ε), (2)
in the sense (4) .
2 Results and proofs
In this section we investigate solvability and convergence properties for the problem (5) and
deduce the corresponding results for the Riemann problems (1ε), (2) .
Proposition 2 For all ε > 0 , u− < u+ and f ∈ C[u−, u+] there exists a unique solution
to the problem (5) .
Proof: Let argue by reductio ad absurdum in order to prove the uniqueness of a solution to
(5) . Assume Φ,Ψ are two different solutions, and c be a point of (positive) local maximum
of (Φ − Ψ) on [u−, u+] . In fact, in case Φ(c) < f(c) one could get a contradiction
from the standard maximum principle. In the general case, one can find ∆ > 0 such that
[c, c + ∆] ⊂ [c, b) and Φ ≥ εf−Φ
> εf−Ψ
= Ψ in the sense of measures on [c, c + ∆] .
Indeed, one has Ψ(c) < Φ(c) ≤ f(c) , so that it can be assumed that Ψ ∈ C2[c, c+∆] and
satisfies (5) with equality on [c, c + ∆] . Moreover, the assumption of maximality above,
together with the convexity of Φ , imply that there exists Φ(c) = Ψ(c) . It follows that
Φ(c+ δ ± 0) > Ψ(c+ δ ± 0) for all δ ∈ (0,∆) , which contradicts to the choice of c .
In order to prove the existence, introduce the penalized problem∗
(7)
Φn(u) = Gn(u,Φn(u)) =
εf(u)−Φn(u)
∧ n, n = 1, 2, ...
Φn(u±) = f(u±), Φn ∈ C2[u−, u+].
Since Gn(u,Φ) is continuous on u and Φ is bounded, there exists a solution of (7) .
The maximum principle is verified for equations of type (7) , because Gn(u,Φ) is in-
creasing in Φ . Set G =√ε(u− u−)(u+ − u) and denote by F the convex hull of f on
[u−, u+] . One has (F − G) ≥ −G ≥ εG≥ ε
f−(F−G), so that (F − G) is a subsolution of
the problem (7) corresponding to G∞ = εf(u)−Φ
. One finds for n ≥ m that G∞(u,Φ) ≥≥ Gn(u,Φ) ≥ Gm(u,Φ) ; by the maximum principle, it follows that Φm(u) ≥ Φn(u) ≥≥ F (u)−G(u) on [u−, u+] .
∗Let a ∧ b denote minb,maxa, 0 for a, b ∈ IR .
1.I.2. Results and proofs 41
Thus Φn(u) ↓ Φ(u) ∈ IR for all u ∈ [u−, u+] ; in addition, Φ(u±∓0) = Φ(u±) = f(u±)
so that Φ ∈ C[u−, u+] and Φ is convex. Moreover, Gn(u,Φn(u)) tends to εf(u)−Φ(u)
∈ IR+
on [u−, u+] . Take a test function ϕ ∈ C∞0 (u−, u+) such that ϕ ≥ 0 ; by (7) and the
Fatou lemma, one has
(8)
u+∫
u−
ϕΦ(u) du = limn→∞
u+∫
u−
ϕΦn(u) du =
= limn→∞
u+∫
u−
ϕGn(u,Φn(u)) du ≥u+∫
u−
ϕε
f(u)− Φ(u)du.
Therefore Φ ≥ εf−Φ
in the sense of measures on (u−, u+) , andε
f−Φ∈ L
loc1 (u−, u+) .
Now take a test function ϕ ∈ C∞0 (u−, u+) with suppϕ ⊂ u|Φ < f . In this
case (8) becomes an equality, because there exists N = N(ϕ) such that for all n ≥ N ,
Gn(u,Φn(u)) ≤ GN (u,ΦN(u)) on suppϕ . Since f − Φ = 0 on [u−, u+] \ u|Φ < f ,one has (f − Φ)
(Φ− ε
f−Φ
)= 0 in the sense of measures on (u−, u+) . ⋄
Remark 2 It is easy to show that for a Lipschitz continuous flux function f the solution of
(5) is a classical solution to the equation Φ = εf−Φ
on (u−, u+) . Nevertheless, it can be
shown using the maximum principle that for f(u) =√u and for all interval (u−, u+) ∋ 0
sufficiently small, the derivative of the solution to (5) has a positive jump at 0 . This jump
corresponds to an interval of ξ where the solution U(·) of (1ε), (2) is constant.
Propositions 1 and 2 yield the following result.
Theorem 1 For all ε > 0 , u− < u+ and f ∈ C[u−, u+] there exists a unique solution Uε
to the problem (1ε), (2) in the sense of Definition 2. This solution is given by the formula
analogous to (3) :
Uε(t, x) = Uε(x/t) = ∂/∂x maxu−≤v≤u+
(xv − tΦε(v)),
where Φε(·) is the unique solution of (5) .
Since we are interested in passing to the limit as ε → 0 , let introduce the subscript ε in
the notation for solutions of (4) and (5) . We need the following two lemmae.
Lemma 2 Let F (·) be the convex hull of f(·) on [u−, u+] . Then Φε(·) converge to F (·)uniformly on [u−, u+] as ε → +0 .
Proof: The proof is based upon a kind of maximum principle argument.
First note that Φε ≤ F . On the other hand, for all α > 0 there exists a function
G ∈ C2[u−, u+] such that 0<F−G<α and G≥C(α)>0 on [u−, u+] . Let c(ε, α) be a
point of global maximum of G−Φε on [u−, u+] ; assume that this maximum is positive. Then
42 Riemann problem for Scalar Conservation Laws
Φε(c) ≤ G(c) < F (c) ≤ f(c) , so that there exists Φε(c) ≥ G(c) ≥ C(α) . Consequently,ε
f(c)−Φε(c)≥ C(α) , and for all ε small enough one has G−Φε < α on [u−, u+] . It follows
that F − Φε < 2α on [u−, u+] in this case; besides, this last inequality is evident in case
the maximum is nonpositive, whence the claim of the lemma. ⋄
Lemma 3 Let F n(·) , n = 1, 2, ... be a sequence of convex functions converging to F 0(·)on [u−, u+] . Then the sequence Un(·) tends to U0(·) at the points ξ of continuity of
U0(·) , where Un(·) , n = 0, 1, ... are the functions constructed by the formula (3) applied
to the functions F n .
Lemma 3 is a corollary of the Fenchel formula and general theorems of the convex analysis
and basic probability theory. An elementary proof can be found in [BA1].
Finally, we establish the relation between the problem (10), (2) and the self-similar viscosity
regularized problems (1ε), (2) , ε > 0 .
Theorem 2 Let Uε be the solution to the problem (1ε), (2) in the sense of Definition 2
(which exists and is unique, due to Theorem 1). Then Uε converge a.e. on Π+ to the
function
U(t, x) = U(x/t) = ∂/∂x maxu−≤v≤u+
(xv − tF (v)) ≡ ∂/∂x maxu−≤v≤u+
(xv − tf(v))
as ε→ +0 , and U is a generalized entropy solution of the problem (10), (2) .
Proof: The convergence of Uε to U follows readily from Lemmae 2,3 and the Fenchel
formula. It is easy to see that U(t, x) → u± as x/t → ±∞ , which implies that (i) of
Definition 1 holds. Moreover, U is a limit of viscous approximations of (10), (2) (i.e., a
wave-fan admissible weak solution). Using Kruzhkov’s techniques (cf. [K69a, K69b, K70a]),
one easily deduces that U satisfies (i) of Definition 1 as well. ⋄
Remark 3 As it is underlined in [Tz96], the wave fan admissibility criterion is different from
usual admissibility conditions for the Riemann problem in that it penalizes the whole fan of
shocks and rarefactions in the solution and not the shocks one by one. In this sense, it is
closer to the global criteria such as the Kruzhkov entropy admissibility criterion (Definition 1)
or the Dafermos entropy rate admissibility criterion (cf. [D73b]).
In piecewise continuous solutions of scalar conservation laws with continuous flux function,
the Kruzhkov criterion still decides on admissibility of each shock in a solution separately; it
induces the conditions
(9) S(u) := sign (ur−ul)s(u−ul)− (f(u)−f(ul)) ≤ 0 for all u between ul and ur,
and S(ur) = 0 if the shock joins ul at the left to ur at the right and propagates with
the speed s . This property is directly motivated by the wave fan admissibility criterion for
1.I.2. Results and proofs 43
the Riemann problem, on account of the explicit representation for the solution. At the same
time, motivation by the travelling waves condition, which is the most common outcome of the
self-similar viscosity approach, becomes more delicate when f is not Lipschitz continuous.
Indeed, this condition requires that an admissible shock could be approximated by solutions
Uε to the equation
(10) Ut + f(U)x = εUxx
of the form Uε(t, x) = Uε((x − st)/ε) † (in case of multiple inflexion points, one admits
also that it be a limit of such shocks with respect to infinitesimal perturbations of f or/and
ul, ur ). Such approximation is possible if and only if there exists a solution to the problem
(11)d
dζU(ζ) = S(U(ζ)), U(−∞) = ul, U(+∞) = ur.
In case f is Lipschitz continuous, this yields the condition (9) with strict inequality, by virtue
of uniqueness for the ODE Cauchy problem. In case of f continuous, analysis of stationary
points of the equation (11) shows that one still can deduce (9) from the travelling waves
condition.
The relation between the usual and self-similar viscosity limits for the Riemann problem
(10), (2) becomes less clear in absence of regularity of f . Kalashnikov in [Ka59] proves the
equivalence, starting from the uniqueness of a self-similar viscous limit and using the maximum
principle to compare primitives of solutions to (1ε) and (10) . In absence of regularity of
f , this comparison becomes delicate†. A reason for the equivalence remains the uniqueness
theory for generalized entropy solutions (e.g., cf. [KPA75] and Chapter 1.IV).
In conclusion, the wave fan admissibility criterion directly selects a unique solution to the
Riemann problem for a scalar conservation law with continuous flux function. As it is shown
in Chapters 1.II and 1.III, the same is true for the p-system and the corresponding system in
Eulerian coordinates, even in the case vaccuum appears; for the p-system, the hyperbolicity
condition can be omitted.
†Clearly, both travelling waves and self-similar viscosity approaches to the Riemann problem are motivated
by introducing the vanishing viscosity of type (10), which is the original idea came from the fluid mechanics
(cf. [Ray10]). Even if the rigorous mathematical study of discontinuous solutions of conservation laws had
started from such a description for Burgers’ equation, given by E.Hopf in his pioneering work [H50], the limits
of solutions of (10) remain difficult to describe directly, especially in case of systems. The recent success in
proving uniqueness for the Glimm scheme and the wave-tracking algorithm, due to A.Bressan and collaborators
(cf. [Br99] for a survey of results), does not promise the uniqueness of solutions that are limits of vanishing
viscosity unless uniform BV estimates on viscous solutions are obtained.
On the other hand, approximation by travelling waves or self-similar viscosity permits to pursue the analysis
within the field of ordinary differential equations, where a variety of tools apply. From the physical viewpoint,
both seem acceptable but not natural.
†Maximum principle is an argument of the same order as the monotony of Uε in Lemma 1. While the
assertion of Lemma 1 is evident when f is smooth, the author was unable to establish it in the general case
by a simpler method than that exposed in the proof.
CHAPTER 1.II
The Riemann Problem for p-System
with Continuous Flux Function†
Introduction
Consider the Riemann problem for a so-called p-system, i.e. the initial-value problemUt − Vx = 0
Vt − f(U)x = 0, (U, V ) : (t, x) ∈ IR+ × IR 7→ IR2; (1)
U(0, x) =
u+, x > 0
u−, x < 0, V (0, x) =
v+, x > 0
v−, x < 0u±, v± ∈ IR. (2)
The flux function f : IR 7→ IR is assumed to be continuous and strictly increasing (except in
Section 5, where the monotony assumption is relaxed).
In the case of piecewise smooth flux function the problem (1),(2) was treated by L.Leibovich,
[Le74] (cf. also [ChHs] and references therein). By analyzing the wave curves on the plane
(u, v) it has been shown that a self-similar distribution solution that is consistent with a
certain admissibility criterion (cf. B.Wendroff, [We72]; also I.Gelfand, [G59] and S.Kruzhkov,
[K70b] for the original idea carried out in the case of scalar conservation laws) may be explicitly
constructed through convex and concave hulls of the flux function f . It has been noticed by
C.Dafermos in [D74] that the same solution satisfies the wave fan admissibility criterion, i.e.,
it can be obtained as limit of self-similar viscous approximations as viscosity goes to 0 . Here
we follow this last idea.
Let introduce some notation. For given [a, b] ⊂ IR and f : u ∈ [a, b] 7→ IR continuous,
the convex hull of f on [a, b] is the function u ∈ [a, b] 7→ supφ(u) | φ is convex and φ ≤
†This chapter extends the author’s graduate paper [BA0] written at the Chair of Differential Equations
under the supervision of S.N.Kruzhkov. The contents of this chapter, excluding Sections 4 and 5, will be
published in [BA3].
46 Riemann problem for p-Systems
f on [a, b]. Respectively, the concave hull of f on [a, b] is the function u ∈ [a, b] 7→
infφ(u) | φ is concave and φ ≥ f on [a, b]
. Take u0 in IR ; by F+(·; u0) denote the
convex hull of f on [u0, u+] if u0 ≤ u+ , and the concave hull of f on [u+, u0] if
u0 ≥ u+ . Replacing u+ by u− , define F−(·; u0) in the same way. Let shorten F±(·; u0)to F± when no confusion can arise.
Since f is strictly increasing, the inverse of dF+
du, denoted by
[dF+
du
]−1
, is well defined in
the graph sense as function from [0,+∞) to [u0, u+] if u0 < u+ (respectively, to [u+, u0]
if u0 > u+ ). In the case u0 = u+ let[dF+
du
]−1
mean the function on [0,+∞) identically
equal to u0 . With the same notation for F−, u− in place of F+, u+ and F± standing fordF±
du, which are non-negative, the self-similar solution of the problem (1),(2) constructed in
[Le74] may be written as
U(t, x) =
[F+(·; u0)
]−1
(x2/t2), x ≥ 0[F−(·; u0)
]−1
(x2/t2), x ≤ 0, (3)
V (t, x) = v− −∫ x/t
−∞
ζdU(ζ), (4)
dU(ζ) being regarded as measure; and, for a bijective flux function f , the value u0 is
uniquely determined by
v− − v+ =
∫ u+
u0
√F+(u; u0)du+
∫ u−
u0
√F−(u; u0)du. (5)
In the case of bijective locally Lipschitz continuous flux function f , the same formulae (3)-
(5) were obtained by P.Krejcı, I.Straskraba ([KrSt93]) for the unique solution to satisfy their
“maximal dissipation” condition. This solution was also shown to be the unique a.e-limit as
ε→ 0 of solutions to Riemann problem for the p-system regularized by means of infinitesimal
parameter ε > 0 , introduced into the flux function f , and the viscosity
(0
εtVxx
).
In this chapter a refinement of these results is presented. The techniques employed are
those used by the author while treating the Riemann problem for a scalar conservation law with
continuous flux function (cf. Chapter 1.I and [BA1],[BA2]). In the general case of continuous
strictly increasing flux function f , the Riemann problem (2) for the p-system (1) and the
regularized system
Ut − Vx = 0
Vt − f(U)x = εtVxx(6)
are treated. The main result is the following theorem:
1.II.1. Restatement of the problem 47
Theorem 1 Suppose f : IR → IR is increasing and bijective. Then for all u±, v± ∈ IR ,
ε > 0 there exists a unique bounded self-similar distribution solution (Uε, V ε) of the problem
(6),(2).
Besides, as ε ↓ 0 , (Uε, V ε)(ξ) → (U, V )(ξ) a.e. on IR , where (U, V ) is given by the
formulae (3)-(5), so that (U, V ) is a self-similar distribution solution of the problem (1),(2).
The bijectivity condition is only needed for the existence of solutions and cannot be omitted
(see Remark 7.6 in [KrSt93]), though it can be relaxed (see Remark 2 in Section 3).
The chapter is organized as follows. In the first section the problem (6),(2) is reduced
to a pair of boundary-value problems for a second-order ordinary differential equation on the
domains (minu0, u±,maxu0, u±) ; u0 is a priori unknown and satisfies an additional
algebraic equation. In Section 2 existence, uniqueness and convergence (as ε → 0 ) results
are obtained for the ODE problem stated in Section 1, with u0 ∈ IR fixed. Then it is shown in
Section 3 that u0 is in fact uniquely determined by the flux function f , ε , and the Riemann
data u±, v± ; finally, Theorem 1 above is proved†.
1 Restatement of the problem
We start by fixing ε > 0 . Consider the problem (6),(2) in the class of bounded distribution
solutions (U, V ) of (6) such that (U, V )(t, ·) tends to (U, V )(0, ·) in L1loc(IR) × L1
loc(IR)
as t tends to +0 essentially. Moreover, since both the initial data (2) and the system (6)
are invariant under the transformations (t, x) → (kt, kx) with k in IR (here is the reason
to introduce the viscosity with factor t ), it is natural to seek for self-similar solutions, i.e.
(U, V ) depending solely on the ratio x/t . By abuse of notation, let write (U, V )(t, x) =
(U, V )(x/t) . Let ξ denote x/t and use U ′, V ′ for dU/dξ, dV/dξ and so on.
Lemma 1 A pair of bounded functions (U, V ) : ξ ∈ IR 7→ IR2 is a self-similar distribution
solution of (6),(2) if and only if U, V, ξU ′ and V ′ are continuous on IR , the equations
εξU ′(ξ) = −∫ ξ
0
ζ2U ′(ζ)dζ + f(U(ξ)) + C (7)
V (ξ) = −∫ ξ
0
ζU ′(ζ)dζ +K (8)
are fulfilled with some constants C,K, and also
U(±∞) = u±, V (±∞) = v±. (9)
Besides, there exist ξ± in IR± , ξ− ≤ ξ+ , such that U, V are strictly monotone on each
of (−∞, ξ−) , (ξ+,+∞) , with U ′ 6= 0 , and U, V are constant on (ξ−, ξ+) .
†In Section 4 we present some comments on the results obtained in Section 3. Section 5 is devoted to the
case where the monotonicity assumption on f is relaxed, which gives rise to a hyperbolic-elliptic problem.
48 Riemann problem for p-Systems
Proof: Let (U, V ) be bounded self-similar distribution solution of the system (6). Then
−ξU ′−V ′ = 0 and −ξV ′−f(U)′ = εV ′′ in D′(IR) ; therefore[ξ2U−f(U)+εξU ′
]′= 2ξU
in D′(IR) . Since U ∈ L∞(IR) , it follows that
ξ2U − f(U) + εξU ′ =
∫ ξ
0
2ζU(ζ)dζ + C ∈ C(IR) (10)
with some C in IR . Hence one deduce consecutively that ξU ′ ∈ L∞loc(IR) , U ∈ C(IR\0)
and finally, U ∈ C1(IR\0) . Thus for all ξ 6= 0 (7) holds.
Now let prove the monotony property stated. For (ξ−, ξ+) take the largest interval in IR
containing ξ = 0 such that U = U(0) on (ξ−, ξ+) . For instance, let ξ+ be finite and
therefore U not constant on (0,+∞) ; suppose U is not strictly monotone on (ξ+,+∞) .
Since U ′ ∈ C(ξ+,+∞) , it follows that there exists c > ξ+ such that U ′(c) = 0 and
U ′ is non-zero in some left neighbourhood of c . For instance, assume U ′ > 0 in this
neighbourhood. Clearly, there exists a sequence ξn ⊂ IR increasing to c such that for all
n ∈ IN the maximum of U ′ on [ξn, c] is attained at the point ξn . Since f is increasing,
it follows that f(U(ξn)) < f(U(c)) . Take (7) at the points ξ = ξn and ξ = c ; subtraction
yields
εξnU′(ξn)− εc · 0 ≤
∫ c
ξn
ζ2U ′(ζ)dζ + f(U(ξn))− f(U(c)) ≤ U ′(ξn)
∫ c
ξn
ζ2dζ.
As n→ ∞ , one deduces that ε ≤ 0 , which is impossible.
Thus U , and consequently V , are indeed monotone on (−∞, 0) and (0,+∞) ; there-
fore there exist U(±0) = limξ→±0 U(ξ) . Hence by (10) there exist limξ→±0 ξU′(ξ) , which
are necessarily zero since U ∈ L∞(IR) . Thus (10) yields f(U(+0)) = f(U(−0)) , so that
U ∈ C(IR) . Consequently, ξU ′ ∈ C(IR) , V ′ ∈ C(IR) , and V ∈ C(IR) . It follows that
(7),(8) hold for all ξ in IR .
The converse assertion, i.e. that (7),(8) imply (6) in the distribution sense, is trivial.
Finally, since U and V are shown to be monotone on IR± whenever (7),(8) hold, it is
evident that (9) is fulfilled if and only if self-similar U, V satisfy (2) in L1loc -sense as t→ 0
essentially. ⋄
Let use this result to obtain another characterisation of self-similar solutions to (6),(2).
The idea is to seek for solutions of the same form as in formulae (3)-(5), substituting F± by
appropriate functions depending on ε . One thus has to “inverse” (3)-(5).
Set u0 := U(0) and consider (7) separately on (−∞, ξ−) , (ξ−, ξ+) , and (ξ+,+∞) ,
where ξ± are defined in Lemma 1. Assume u0 6= u− , u0 6= u+ . Let introduce the
notation I(a, b) for the interval between a and b in IR . One has U(ξ) = u0 for all
ξ ∈ (ξ−, ξ+) ; besides, the inverse functions U−1+ : I(u0, u+) 7→ (ξ+,+∞) and U−1
− :
1.II.1. Restatement of the problem 49
I(u0, u−) 7→ (−∞, ξ−) are well defined. For all u ∈ I(u0, u+) (respectively, u ∈ I(u0, u−) )
set
Φε+(u; u0) :=
∫ u
u0
(U−1+ (w)
)2dw − C
(resp., Φε
−(u; u0) :=
∫ u
u0
(U−1− (w)
)2dw − C
)(11)
with C taken from (7). The shortened notation Φ±(u) will be used for Φε±(u; u0) whenever
ε, u0 are fixed. Now (7) can be rewritten as εξU ′(ξ) = f(U(ξ)) − Φ±(U(ξ)) for ξ ∈I(ξ±,±∞) . The reasoning in the proof of Lemma 1 shows that U is not only monotone,
but also U ′ is different from 0 outside of [ξ−, ξ+] . It follows that for all u in I(a, b) ,
where a = u0 , b = u+ (resp., for all u in I(a, b) , where a = u0 , b = u− ), the function
Φ+ (resp., Φ− ) is twice differentiable and satisfies the equation
Φ(u) =2εΦ(u)
f(u)− Φ(u), with Φ(u) > 0 and Φ(u) · (b− a) > 0. (12)
Hence Φ+ < f ( Φ+ > f ) if u0 < u+ (if u0 > u+ ), and the same for Φ− , u− in place
of Φ+ , u+ .
Note that one can extend the functions Φ+ , Φ− to be continuous on I(u0, u+) , I(u0, u−)
respectively, and in this case one has
Φ+(u0) = f(u0), Φ+(u+) = f(u+)(resp., Φ−(u0) = f(u0), Φ−(u−) = f(u−)
). (13)
Indeed, one gets Φ±(u0) = f(u0) directly from (11) and (7). Besides, for ξ ∈ IR± , εξU ′(ξ)
is equal to f(U(ξ))−Φ±(U(ξ)) , which has finite limits as ξ → ±∞ because U(±∞) = u±
and Φ± are convex and bounded on I(u0, u±) . The limits of εξU ′(ξ) cannot be non-zero
since U is bounded, thus one naturally assign Φ±(u±) := f(u±) .
Now from (8)-(11) it follows that
v− − v+ =
∫ u+
u0
√Φε
+(u; u0)du+
∫ u−
u0
√Φε
−(u; u0)du. (14)
Note that in the case u0 = u+ ( u0 = u− ), (12)-(14) formally make sense, with Φ+
defined at u = u0 = u+ by f(u+) (resp., with Φ− defined at u = u0 = u− by f(u−) ).
Finally, the reasoning above is inversible. More presisely, for given u0 ∈ IR and Φε±(·; u0) ∈
C2(I(u0, u±)) ∩ C(I(u0, u±)) such that (12)-(14) hold, define U, V by
U(ξ) =
[Φε
+(·; u0)]−1
(ξ2), ξ ≥ 0[Φε
−(·; u0)]−1
(ξ2), ξ ≤ 0(15)
V (ξ) = v− −∫ ξ
−∞
ζdU(ζ), (16)
with [Φε+(·; u0)]−1 (and [Φε
−(·; u0)]−1 ) taken in the graph sense and equal to u+ (to u− )
identically whenever u0 = u+ ( u0 = u− ). Then (U, V ) satisfy (7)-(9). Indeed, U is
50 Riemann problem for p-Systems
continuous, Φε+(u0; u0) = Φε
−(u0; u0) , and the equation εξU ′(ξ) = f(U(ξ))−Φε±(U(ξ); u0)
holds for all ξ ∈ IR± . Hence ξU ′ ∈ C(IR) and (7) is true. Therefore V ′, V are continuous
and (8),(9) are easily checked.
We collect the results obtained above in the following proposition:
Proposition 1 Let ε, f, u±, v± be fixed. Formulae (15),(16) provide a one-to-one corre-
spondence between the sets A and B defined by
A :=(u0,Φ±(·)
)| u0 ∈ IR, Φ± : I(u0, u±) 7→ IR, Φ± ∈ C2(I(u0, u±)) ∩ C(I(u0, u±))
and (12)− (14) hold
B :=(U, V ) | (U, V ) is a bounded self-similar distribution solution of (6), (2)
In fact, it will be shown in Section 3 that A and thus B are one-element or empty sets.
The resemblance of formulae (3),(4),(5) and (15),(16),(14) permits to get the convergence
result of Theorem 1 if one has convergence of Φε± to F± as ε → 0 .
2 The problem (12),(13) with fixed domain
Let fix a, b ∈ IR and consider the equation (12) on the interval I(a, b) , with the boundary
conditions as in (13). For instance, suppose a ≤ b .
Proposition 2 For all continuous strictly increasing f , ε > 0 , and a, b ∈ IR there exists a
unique Φ in C2(I(a, b)) ∩ C(I(a, b)) satisfying (12) such that Φ(a) = f(a) and Φ(b) =
f(b) .
For f and [a, b] fixed, let Φε denote the function Φ from Proposition 2 corresponding
to ε , ε > 0 .
Proposition 3 With the notation above, Φε converge in C[a, b] , as ε → 0 , to the convex
hull F of the function f on the segment [a, b] .
Remark 1 In the case a ≥ b , the corresponding limit is the concave hull of f on [b, a] .
The following two assertions will be repeatedly used in the proofs in Sections 2,3:
Lemma 2 (Maximum Principle) Let Φ,Ψ ∈ C2(a, b) ∩ C[a, b] and satisfy, for all u ∈
(a, b) , the equations Φ(u) = G(u,Φ(u), Φ(u)) and Ψ(u) = H(u,Ψ(u), Ψ(u)) , respectively,
with G,H : (a, b)× IR × (0,+∞) 7→ (0,+∞] .
a) Assume that G(u, z, w) < H(u, ζ, w) for all u ∈ (a, b) such that Φ(u) < Ψ(u)
and all z, ζ, w such that z < ζ . Then Φ ≥ Ψ on [a, b] whenever Φ(a) ≥ Ψ(a) and
Φ(b) ≥ Ψ(b) .
b) Assume that G(u, z, w) ≡ H(u, z, w) , increases in w and strictly increases in z ; let
Φ(a) = Ψ(a) or Φ(b) = Ψ(b) . Then (Φ−Ψ) is monotone on [a, b] .
1.II.2. Problem (12),(13) with fixed domain 51
Proof: The proof is straightforward. ⋄
Lemma 3 Let functions F, Fn , n ∈ IN , be continuous and convex (or concave) on [a, b] .
Assume that Fn(u) converge to F (u) for all u ∈ [a, b] . Then this convergence is uniform
on all [c, d] ⊂ (a, b) and
a) Fn converge to F a.e. on [a.b] ;
b) if Fn, F are increasing, then
∫ b
a
√Fn(u)du converge to
∫ b
a
√F (u)du ;
c) let[F]−1
,[Fn
]−1
denote the graph inverse functions of F, Fn respectively; then[Fn
]−1
(ξ) tends to[F]−1
(ξ) for all ξ such that[F]−1
is continuous at the point ξ .
Proof: An elementary proof of a),c) is given in [BA1]. Besides, the assumptions of the
Lemma imply that for all δ > 0 Fn are bounded uniformly in n ∈ IN for u ∈ [a+ δ, b− δ] .Since, in addition,
∣∣∣∣∫ a+δ
a
√Fn(u)du+
∫ b
b−δ
√Fn(u)du
∣∣∣∣→ 0 uniformly in n ∈ IN as δ →0 , the conclusion b) follows from the Lebesgue Theorem. ⋄
Proof of Proposition 2: There is nothing to prove if a = b ; let a < b . Consider the
penalized problem
Φ(u) = Gn(u,Φ(u), Φ(u)) :=
2εΦ(u)
f(u)−Φ(u), if this value is in (0, n)
n, otherwise, Φ(u) > 0 (17)
for all u ∈ [a, b] . Since Gn is continuous in all variables and bounded, the existence of
solution follows for arbitrary boundary data such that Φ(a) < Φ(b) ; in particular, a solution
Φn exists such that Φn(a) = f(a) , Φn(b) = f(b) . The Maximum Principle yields that Φn
decrease to some convex non-decreasing function Φ on [a, b] as n→ ∞ .
Further, there exists a solution Ψ of (12) on [a, b] with any assigned value of Ψ(a) less
than f(a) , or any assigned value of Ψ(b) less than f(b) . In fact, in the first case one
takes Ψ(u) ≡ Ψ(a) ; in the second case there exists a solution on the whole of [a, b] to
the equation (12) with the Cauchy data Ψ(b) (fixed) and Ψ(b) sufficiently large. By the
Maximum Principle Φn ≥ Ψ on [a, b] ; therefore Φ(a + 0) = f(a) and Φ(b − 0) = f(b) .
Consequently Φ is continuous on [a, b] .
Now if for all [c, d] ⊂ (a, b) there exists m0 > 0 such that f −Φ ≥ m0 on [c, d] , then
the functions Gn(u,Φn(u), Φn(u)) are bounded uniformly in n ∈ IN for u ∈ [c, d] ; indeed,
on [c, d] , by convexity, Φn are uniformly bounded and Φn converge to Φ uniformly, so
that 2εΦn
f−Φn≤ M(c, d) for all n large enough. Hence it will follow by Lemma 3a) and the
Lebesgue Theorem that Φ(u) = 2εΦ(u)f(u)−Φ(u)
for all u ∈ [c, d] , and consequently Φ ∈ C2[c, d] .
Thus the existence of solution to problem (12),(13) will be shown.
First let show that Φ(u ± 0) > 0 for all u > a . It suffices to prove that u = a ,
where u := supu ∈ [a, b] |Φ(u) = f(a)
. Note that u < b since Φ(b) = f(b) > f(a) .
52 Riemann problem for p-Systems
Assume u > a ; by the Lebesgue Theorem Φ = 2εΦf−Φ
in some neighbourhood of u . Since
Φ(u − 0) = 0 , by the uniqueness theorem for the Cauchy problem Φ is constant in this
neighbourhood. Therefore necessarily u = b , which is impossible.
Further, by Lemma 3a), (17), and the Fatou Lemma one has 2εΦf−Φ
∈ L1loc(a, b) . Hence
Φ ≤ f and 2εΦf−Φ
≤ Φ on (a, b) in measure sense. Now take [c, d] ∈ (a, b) and u ∈ [c, d] ;
set m := f(u) − Φ(u) ≥ 0 . Set A := Φ(a+c2
− 0) > 0 , B := Φ(d − 0) > 0 . For all
u ∈ [a+c2, u] , f(u)− Φ(u) ≤ m+ B(u− u) and Φ(u± 0) ≥ A since Φ is convex and f
increasing. Hence
B −A ≥∫ u
a+c2
Φdu ≥∫ u
a+c2
2εΦ(u)
f(u)− Φ(u)du ≥
∫ u
a+c2
2εA
m+B(u− u)du = K1 −K2 lnm,
with some positive constants K1, K2 depending only on c, d . Thus m ≥ m0(c, d) > 0 and
the proof of existence is complete.
The uniqueness is clear from the Maximum Principle for solutions of (12). ⋄
Proof of Proposition 3: Let a < b ; take α > 0 and a barrier function Ψα such that
α/2 ≤ F − Ψα ≤ α and Ψα ≥ m(α) > 0 on [a, b] . Such a function can be constructed
through the Weierstrass Theorem.
By the Maximum Principle Φε increase as ε decrease. Therefore there exists [c, d]
inside (a, b) such that for all ε in (0, 1) Φε ≥ Ψα on [a, b] \ [c, d] . It follows thatu |Φε(u) < Ψα(u)
⊂ [c, d] and thus Φε ≤ M(α) on this set uniformly in ε . Now
for all ε less than α·m(α)2M(α)
one may apply the Maximum Principle to Φε and Ψα , hence
0 ≤ F − Φε ≤ α for all ε small enough. ⋄
3 Solutions of the problem (6),(2) and the proof of Theorem
1
Proposition 2 above implies that for all f , ε , u± fixed, for all u0 ∈ IR there exist unique
Φε+(·; u0) and Φε
−(·; u0) satisfying (12),(13); thus by Proposition 1, for an arbitrary v− in
IR and v+ obtained from (14), (U, V ) provided by (15),(16) is a self-similar solution to the
Riemann problem (6),(2). Now since not u0 but v± are given by (2), one needs to find u0
in IR such that (14) holds with these assigned values of v± .
Proposition 4 a) Assume f(±∞) = ±∞ . Then for all u±, v± ∈ IR , ε > 0 there exists a
unique u0 such that (14) holds, with Φε+ , Φε
− the (unique) solutions to (12),(13).
b) Assume f ∈ W 11 locally in IR and
∫ ±∞
0
√f(u)du = ±∞ . Then for all u±, v± ∈ IR
and ε < ε0 = ε0(u±, v+ − v−) there exists a unique u0 such that (14) holds, with the same
Φε± .
1.II.3. Problem (6),(2) and proof of Theorem 1 53
Let F±(·; u0) be, as in the Introduction, the convex (concave) hulls of f on I(u0, u±)
according to the sign of (u± − u0) . Set
∆ε±(u0) :=
∫ u+
u0
√Φε
±(u; u0)du, ∆0±(u0) :=
∫ u+
u0
√F±(u; u0)du.
It will be convenient to extend Φε±(·; u0) , F±(·; u0) to continuous functions on IR by setting
each of them constant on (−∞,minu0, u±] and [maxu0, u±,+∞) . In the lemma below
a few facts needed for the proofs of Proposition 4 and Theorem 1 are stated.
Lemma 4 With the notation above, and u0 running through IR , the following properties
hold.
a) For all u ∈ IR and ε > 0 , u0 7→ Φε±(u; u0) do not decrease; nor do u0 7→ F±(u; u0) .
b) For all u ∈ IR and ε > 0 , u0 7→ sign(u±−u0)Φε±(u; u0) do not increase; nor do
u0 7→ sign(u±−u0)F±(u; u0) .
c) For all ε > 0 the maps u0 7→ Φε±(·; u0) are continuous for the L∞(IR) topology; so
do u0 7→ F±(·; u0) .d) For all ε ≥ 0 , u0 7→ ∆ε
±(u0) are continuous and strictly decreasing.
Proof: Combining the continuity and monotony of f with a),b) of the Maximum Principle
for solutions of (12),(13), one gets a)-c) for Φε± . The same assertions for F± follow now
from Proposition 3 and Lemma 3a); they can also be easily derived from the definition of
convex hull. Finally, d) results from c), Lemma 3b), b) and the strict monotony of f . ⋄Proof of Proposition 4: a) By Lemma 4d), it suffices to prove that ∆ε
±(±∞) = ∓∞ .
Assume the contrary, for instance that ∆ε+(−∞) =M < +∞ .
Consider u0 < u+ ; Φε+ is convex, therefore for all u0 there exists c = c(u0) ∈ [u0, u+]
such that Φε+(·; u0) ≥ 1 on [c, u+) and Φε
+(·; u0) ≤ 1 on (u0, c] . By Lemma 4b) c(u0)
increase with u0 . Obviously, for all u0 , M > ∆ε+(u0) ≥ [Φε
+(c; u0) − f(u0)] + [u+ − c] .
Set d := u+ −M ; clearly, c(u0) ≥ d for all u0 . Considering the functions Φε(·; u0) with
u0 → −∞ , one obtains a sequence Ψn such that Ψn satisfy (12) on [d, u+) , Ψn(d) ≤1 , Ψn(u+) = f(u+) , and finally, Ψn(d) → −∞ (this last since Ψn(d) ≤ f(u0) +M →f(−∞) +M = −∞ , as u0 → −∞ ). On the other hand, for n large enough, the unique
solution Ψ to the equation (12) with the Cauchy data Ψ(d) = Ψn(d) , Ψ(d) = 2 is defined
on the whole of [d, u+] , which means that Ψ(u+) < f(u+) . Now by b) of the Maximum
Principle, (Ψ − Ψn) is increasing and thus positive. Hence Ψn(u+) ≤ Ψ(u+) < f(u+) ,
which is a contradiction.
b) Take u0 < u+ . First suppose f ∈ C2[u0, u+] and has a finite number of points
of inflexion; denote by F the corresponding convex hull. The segment [u0, u+] can be
decomposed into the three disjoint sets: M1 :=u | ∃δ > 0 s.t. F ≡ const on (u − δ, u +
δ) ∩ [a, b], M2 :=
u | F (u) = f(u)
\M1 , and M3 finite. Using the Cauchy-Schwarz
inequality on every (c, d) ⊂M1 , one gets
∫ u+
u0
√F (u)du ≡ ∆0
+(u0) ≥∫ u+
u0
√f(u)du .
54 Riemann problem for p-Systems
In the general case, let proceed with the density argument, choosing a sequence fnsuch that fn are increasing and smooth as above, fn → f in C[u0, u+] with
√fn →
√f
in L1[u0, u+] as n → ∞ . Denote the convex hull of fn on [u0, u+] by Fn ; it is easy
to see that ‖Fn − F‖C[u0,u+] ≤ ‖fn − f‖C[u0,u+] → 0 as n → ∞ . By Lemma 4b),
∆0+(u0) = lim
n→∞
∫ u+
u0
√Fn(u)du , so that ∆0
+(u0) ≥∫ u+
u0
√f(u)du in the general case as
well. Thus ∆0+(−∞) = +∞ by the assumption on f .
Now Proposition 3 and Lemma 3b) imply that for given v± in IR , there exists ε0 =
ε0(v+ − v−) such that one has ∆ε+(−L) > |v− − v+| (and in the same way, ∆ε
+(L) <
−|v−− v+| ) for all ε < ε0 whenever L is large enough. Lemma 4d) yields now the required
fact. ⋄Finally, here is the proof of the result announced in the Introduction.
Proof of Theorem 1: The existence and uniqueness of a bounded self-similar distribution
solution to the Riemann problem (6),(2) follow immediately from Propositions 1, 2 and 4.
Now let ε decrease to 0 . Take(uε0,Φ
ε±(·; uε0)
)corresponding to the unique solution
of (6),(2) in the sense of Proposition 1. Take u0 a limit point in IR of uε0ε>0 . Suppose
first uεk0 → u0 ∈ IR , εk → 0 as k → ∞ ; let show that, with the notation as in Lemma
4, Φε+(·; uε0) converge to F+(·; u0) in L∞(IR) . Indeed, take α > 0 ; |uεk0 − u0| < α for
all k large enough. By Proposition 3 and Lemma 4a), there exists ε0 > 0 such that, for all
εk < ε0 , F+(·; u0−α)−α ≤ Φεk+ (·; u0−α) ≤ Φεk
+ (·; uεk0 ) ≤ Φεk+ (·; u0+α) ≤ F+(·; u0+α)+α .
Thus the required result follows from Lemma 4c); clearly, it also holds for Φεk− , F− in place
of Φεk+ , F+ .
Now by Lemma 3b) ∆0+(u0) + ∆0
−(u0) is the limit of ∆εk+ (uεk0 ) + ∆εk
− (uεk0 ) ≡ v− − v+ ;
hence by Lemma 4d), u0 is unique if it is finite. Besides if, for instance, u0 = −∞ , then
for all L ∈ IR , v− − v+ = limεk→0[∆εk+ (uεk0 ) + ∆εk
− (uεk0 )] ≥ ∆0+(L) + ∆0
−(L) by Lemma
4d) and Lemma 3b). It is a contradiction; indeed, it is easy to see that ∆0±(L) → +∞ as
L→ −∞ .
Thus in fact uε0 → u0 as ε → 0 , u0 ∈ IR and (5) holds. Further, let u0 < u± ; the
other cases are similar and those of u0 = u− or u0 = u+ are trivial. For all α > 0 there
exists ε0 = ε0(α) > 0 such that for all ε < ε0 [uε0, u±] ⊂ [u0 − α, u±] . The functions Uε
in the statement of Theorem 1 are given by formula (15), when applied to Φε±(·; u0) with
their natural domains [uε0, u±] . Taking for the domains [u0 − α, u±] , one do not change
Uε(ξ) for ξ 6= 0 and ε < ε0 . The same being valid for U given by (3), one may use the
fact, proved above, that ‖Φε±(·; uε0) − F±(·; u0)‖C[u0−α,u±] → 0 as ε → 0 , and conclude
by Lemma 3c) that Uε(ξ) → U(ξ) for a.a. ξ ∈ IR . Hence it follows by (4),(16) that
V ε → V a.e., so that (U, V ) given by (3)-(5) is the unique a.e.-limit of self-similar bounded
distribution solutions of the problem (6),(2). Thus (U, V ) is a distribution solution of the
Riemann problem (1),(2). ⋄
1.II.4. Comments 55
Remark 2 Note that using b) of Proposition 4 instead of a), one gets a result similar to the
Theorem 1 in the case of f ∈ W 11 locally in IR ,
∫ ±∞
0
√f(u)du = ±∞ ; in fact, the exact
condition is the bijectivity of the functions u0 7→ ∆0±(u0) for continuous strictly increasing
flux function f . Under each of this conditions the existence of bounded self-similar solution
of (6),(2) is guaranteed for all ε < ε0 = ε0(u±, v+ − v−) .
Note
After this paper had been completed, the author had an opportunity to meet Prof. A.E.Tzavaras
and get acquanted with his papers on viscosity limits for the Riemann problem; in particular,
in [Tz95] very close results were obtained for p-systems regularized by viscosity terms of the
form
(0
εt(k(U)Vx)x
), without involving the explicit formulae for the limiting solution.
For results on self-similar viscous limits for general strictly hyperbolic systems of conserva-
tion laws, refer to the survey paper [Tz98] and literature cited therein. Let only note that the
structure of wave fans in self-similar viscous limits remains the same as in the case of scalar
conservation laws ([G59],[K70b]) and in the case of p-systems, where it can be easily observed
through the formulae (3),(4).
On the other hand, Prof. B.Piccoli turned my attention to Riemann solvers for hyperbolic-
elliptic systems (1) (i.e. the case of non-monotone f ). The global explicit Riemann solver
extends to this case (see Krejcı,Straskraba, [KrSt97],[KrSt93]); it can be proved, with the
techniques used here and in [BA1],[BA2], that this solver is the unique limit of self-similar
bounded solutions to the problem (6),(2).
Precise results on hyperbolic-elliptic p-systems and a discussion of other viscosity terms
are given in Section 5 below.
4 Comments
It is possible to treat, in almost the same way, the case of (by no means physical) viscosity
term
(0
εtUxx
). Lemma 1 still holds; moreover, in this case U, V ∈ C1(IR) . For Φε
±(·; u0)
defined by (11), one gets instead of (12) the equation
Φ(u) =2ε√Φ(u)
f(u)− Φ(u)with Φ > 0 and Φ > 0 or Φ < 0;
this problem shares the properties of (12) that were important for us.
Further, the more general viscosity terms
(0
εt(k(U)Vx)x
)or
(0
εt(k(U)Ux)x
), with
k ∈ C(IR; (0,+∞)) , also yield unique approximate solutions, which converge to (U, V ) given
56 Riemann problem for p-Systems
by (3)-(5). The only difference is the factor k(u) in the right-hand side of the equation in
(12).
Finally, the classical example of p-system with f(u) = 1/u , k(u) = 1/u on the domain
u > 0 , given by the isentropic gas dynamics in Lagrangian coordinates, can be included; as
in b) of Proposition 4, the divergence of the integral
∫ √f(u)du at 0 and +∞ guarantees
existence of solution to (6),(2) for all data u± > 0 , v± ∈ IR .
On the contrary, in the case of γ -pressure laws with γ > 1 (i.e. f(u) = 1/uγ ) we
have to impose the restriction v+ − v− <
∫ +∞
u−
√f(u)du +
∫ +∞
u+
√f(u)du in order to
have existence of a bounded self-similar solution to the approximating system. Otherwise, we
have to deal with solutions where u is unbounded, which corresponds to vanishing density
ρ . Still the vaccuum remains invisible in Lagrangian representation†, so that the appropriate
description of such problems has to be given in the Eulerian one. In Chapter 1.III, we will
carry out a thoroughful investigation of the Riemann problem for the system of isentropic
gas dynamics in Eulerian coordinates. Similar results will be obtained for general continuous
pressure laws and arbitrary Riemann data, including those that give rise to vaccuum in the
solutions.
Another extension of techniques applied above is presented in Section 5 below, where we
treat the Riemann problem for hyperbolic-elliptic systems of form (1). Indeed, as it is shown in
†Unless one considers measure-valued solutions. In [Wa87] the equivalence of equations of gas dynamics
in Eulerian and Lagrangian coordinates is shown for a very large class of weak solutions. Moreover, the
transformation between the two representations preserves the class of convex entropies, so that it preserves
the entropy admissibility in the sense of Lax (cf. [K70a, Lax71]).
In particular, the result of [Wa87] applies to the Riemann problem with the two initial states different from
vaccuum. It would be interesting to check whether Wagner’s arguments apply to viscosity regularized systems
(which is, clearly, the case when there is no vaccuum). If yes, the wave-fan admissibility is preserved under the
change of representation. As show the results of Chapter 1.III, the solutions to the Riemann problem (6),(2) are
still ordinary functions, though may be unbounded when the assumption
∫ +∞
1
√f(u)du+
∫ +∞
1
√f(u)du =
+∞ fails. But we have to pass to the limit in the sense of Radon measures, since the Lagrangian equivalent
of the solution (40)-(45) in Chapter 1.III can contain a Dirac mass (a so-called δ -wave).
There is a connection with another interesting question. Namely, one system where δ -waves naturally arise
in solutions to the Riemann problem is the nonstrictly hyperbolic system
ut + (u2/2)x = 0
vt + (uv)x = 0(∗)
considered by K.T.Joseph in [Jo93]. This is the simplest representative of another class of systems where the
idea of the present work could be applied for solving the Riemann problem. Complications will arise of the
same order as indicated above. Still it indicates that there could be an explicit formula for admissible solutions
of the Riemann problem. Not surprisingly, this formula does exist for the system (∗) : it is obtained in [Jo93],
through the viscous regularisation
(εuxx
εvxx
)and the Hopf-Cole-Lax transformation (cf. [H50, Lax57]).
1.II.5. Hyperbolic-elliptic case 57
[KrSt93] (cf. [KrSt97] for a detailed exposition), the monotony of f is not essential; under the
assumption that there exist (finite or infinite) limits f± =: limu→±∞
f(u) and f− < f(u) < f+
for all u , one can justify the appropriate version of formulae (3)-(5) through the maximal
dissipation principle of Krejcı-Straskraba. We give a justification of this formulae by the self-
similar vanishing viscosity approach.
5 The hyperbolic-elliptic case
Let consider the problems (1),(2) and (6),(2) without the monotony assumption on f . In-
stead, we require that
f− < f(u) < f+ for all u, where f− =: lim infu→−∞
f(u), f+ =: lim supu→+∞
f(u). (18)
Self-similar viscosity limits for a class of hyperbolic-elliptic p-systems modelling van der Waals
fluids have been constructed in [Sl89] and [Fan92], in case of the identity viscosity matrix. It
generates solutions that have the same structural properties as the solution (19)-(22) obtained
below as the limit of solutions of (6),(2). In particular, only stationary phase transitions (i.e.,
jumps across elliptic regions) occur in the two cases. This is not always satisfactory; some
comments on this issue are presented in Remark 4 at the end of this section.
As before, let restrict our attention to bounded self-similar solutions. Let (U, V ) be such
a solution to (6),(2) with some ε > 0 . In general, U will not be continuous at ξ = 0 , since
the continuity of f(U) at 0 yields it no more. Neither will U be strictly monotone outside
of a neighbourhood of ξ = 0 . In turn, this implies that the functions Φ±(·) , defined as in
(11), do not necessarily verify (12) in the classical sense. Whence the two modifications we
have to perform, comparing to the case of strictly increasing f .
First, we take f0 = f(U)(0) , which is well defined, for the parameter in our study
of viscous solutions corresponding to given u± and v− . It will be shown that the limits
u0,± := limξ→±0
U(ξ) and the value of v− − v+ are uniquely defined by f0 . The relation with
the problem (12)-(13) on the intervals I(u0,±, u±) , now understood in a weakened sense
similar to (5) in Chapter 1.I, will be established, followed by the existence and uniqueness result
for this problem. Second, in order to overcome the loss of regularity relative to possible sharper
singularities in f , we will follow the ideas carried out in the case of scalar equation (Chapter
1.I). Beyond some minor modifications in other arguments, we have to upgrade Lemma 1 (the
monotony part), Proposition 1 and the Maximum Principle; in turn, this simplifies the proof
of the version of Proposition 2.
Let f0 ∈ (f−, f+) . Under the assumption (18), there exist (unique) u0,± that satisfy
u0,± :=
maxu < u± | f(u) = f0, f0 < f(u±)
u±, f0 = f(u±)
minu > u± | f(u) = f0, f0 > f(u±).
(19)
58 Riemann problem for p-Systems
Denote by F+ = F+(·; u0,+) the convex hull of f on [u0,+, u+] if u0,+ ≤ u+ , and the
concave hull of f on [u+, u0,+] if u0,+ ≥ u+ . Replacing u+ by u− and u0,+ by u0,− ,
define F− = F−(·; u0,−) in the same way. We will prove that the unique viscous limit solution
to (1),(2) is given by the formulae
U(t, x) =
[F+(·; u0,+)
]−1
(x2/t2), x ≥ 0[F−(·; u0,−)
]−1
(x2/t2), x ≤ 0, (20)
V (t, x) = v− −∫ x/t
−∞
ζdU(ζ), (21)
dU(ζ) being regarded as measure; (19) and the formula
v− − v+ =
∫ u+
u0,+
√F+(u; u0,+)du+
∫ u−
u0,−
√F−(u; u0,−)du. (22)
provide a one-to-one correspondence between f0 ∈ (f−, f+) and an interval of admissible
values of v− − v+ . Additional conditions are required in order to have all values of v− − v+
admissible; for example, f± = ±∞ is sufficient (see [KrSt93]).
Let start with
Lemma 5 A pair of bounded functions (U, V ) : ξ ∈ IR 7→ IR2 is a self-similar distribution
solution of (6),(2) if and only if:
· V, V ′, ξU ′ are continuous on IR , U is continuous on IR\0 and admits limits u0,±
as ξ → ±0 such that f(u0,−) = f(u0,+) ;
· the equations
εξU ′(ξ) = −∫ ξ
0
ζ2U ′(ζ)dζ + f(U(ξ)) + C, (23)
V (ξ) = −∫ ξ
0
ζU ′(ζ)dζ +K (24)
are fulfilled with some constants C,K ;
· U(±∞) = u±, V (±∞) = v±. (25)
Besides, U, V are monotone on each of (−∞, 0) , (0,+∞) .
In addition, u0,± are connected to f0 := f(u0,−) = f(u0,+) through (19).
1.II.5. Hyperbolic-elliptic case 59
Proof: Just as in the proof of Lemma 1, we first deduce (23) for ξ > 0 and ξ < 0 .
We would arrive to U ∈ C(IR \ 0) , ξU ′(ξ) ∈ C(IR) with limξ→±0 ξU′(ξ) = 0 , f(U) ∈
C(IR) , and V ∈ C1(IR) , if we only could show that there exist u0,± = limξ→±0U(ξ) .
Again, it follows from the monotony of U on both sides from the singularity point ξ = 0 .
The monotony property can be proved using (23) and the reasoning carried out in Lemma 1
of Chapter 1.I (see also [BA1]). We complete the proof of the equivalence as in Lemma 1.
Further, note that C = −f0 . For instance, let f(u−) < f0 . The formula (23) together
with the monotony of U imply that, first, sign (u−−u0,−) = sign (f(u−)− f(u0,−)) = −1 ;
second, for all u ∈ [u−, u0,−) , f(u) < f0 . These two conditions yield u0,− = minu >
u− | f(u) = f0 . Thus (19) holds for this case; the other cases are similar. ⋄
Let (U, V ) be a bounded self-similar solution of (6),(2). Using the monotony of U on
(−∞, 0) and (0,+∞) , one can inverse it on each of these intervals. The resulting functions
U−1± are defined a.e. on I(u0,±, u±)
† and monotone. For all u ∈ I(u0,±, u±) , set
Φε±(u; u0,±) :=
∫ u
u0,±
(U−1± (w)
)2dw − C (26)
with C taken from (23). We will abrige Φε±(·; u0,±) to Φ±(·) whenever ε and u0,± are
fixed. Now (23) can be rewritten as
εξU ′(ξ) = f(U(ξ))− Φ±(U(ξ)) for ξ ∈ I(0,±∞). (27)
As in Chapter 1.I, set Ω± := u | Φ±(u) = f(u) ≡ u | ∃ξ ∈ IR± such that U(ξ) =
u, U ′(ξ) = 0 . By the Sard lemma, Ω± have the Lebesgue measure 0 . We still obtain
that the equation in (12) is fulfilled outside of Ω± in classical sense. But on the whole of
I(a, b) = I(u0,±, u±) , we only have (12),(13) fulfilled in the following sense:
Φ ∈ C(I(a, b)), Φ is strictly increasing and (b− a)Φ is convex on I(a, b);
(b− a)(f − Φ) ≥ 0 on I(a, b) and Φ(a) = f(a),Φ(b) = f(b);
(b− a)G(·,Φ(·), Φ(·)) ∈ L1loc(I(a, b)) ∩ C(I(a, b); (0,+∞]);
(b− a)(Φ(·)−G(·,Φ(·), Φ(·))
)≥ 0 in the measure sense on I(a, b);
(f(·)− Φ(·))(Φ(·)−G(·,Φ(·), Φ(·))
)= 0 in the measure sense on I(a, b)
(28)
with G : (u, z, w) ∈ I(a, b) × (−∞,+∞) × (0,+∞) 7→ 2εwmaxf(u)−z,0
∈ (0,+∞] . In-
deed, for instance suppose u0,− < u− . Then U−1− is a non-increasing negative function.
Hence Φ is strictly increasing and Φ−(u) = ddu
(U−1−
)2is a non-negative measure on
(u0,−, u−) . By (27), Φ− ≤ f and Φ− can be extended on I(a, b) by continuity, with
Φ−(a) = f(a),Φ−(b) = f(b) . Since 2εΦ−
f−Φ−= Φ− on (u0,−, u−)\Ω− and Ω− is of measure
0 , 2εΦ−
f−Φ−∈ L1
loc(u0,−, u−) . The last two properties in (28) are now evident. Besides, it follows
†As above, for a, b ∈ IR , we denote by I(a, b) the interval (mina, b,maxa, b)
60 Riemann problem for p-Systems
from (19) that f(u0,−) < f(u) for all u ∈ (u0,−, u−] ; this implies, as in the proof of Propo-
sition 2, that Φ− > 0 on (u0,−, u−] . Thus we can write 2εΦ−
f−Φ−∈ C((u0,−, u−]; (0,+∞]) .
Conversely, a pair of functions Φ± verifying (28) on I(u0,±, u−) , where u0,± satisfy
(19), gives rise to a bounded self-similar solution of (6), (2) through the formulae
U(ξ) =
[Φ+(·; u0,+)
]−1
(ξ2), ξ > 0[Φ−(·; u0,−)
]−1
(ξ2), ξ < 0,(29)
V (ξ) = v− −∫ ξ
−∞
ζdU(ζ) (30)
under the following condition that generalizes (14):
v− − v+ =
∫ u+
u0,+
√Φε
+(u; u0,+)du+
∫ u−
u0,−
√Φε
−(u; u0,−)du. (31)
Indeed, (25) is obvious. As above, (28) implies that Φ− > 0 on (u0,−, u−] ; further, as in
Lemma 2 in Chapter 1.I, we deduce (27) for all ξ ∈ IR . Hence (29) yields (23), (24) and the
continuity properties of Lemma 5.
We summarize Lemma 5 and the results above in
Proposition 5 Let ε, f, u±, v± be fixed. Formulae (29),(30) provide a one-to-one corre-
spondence between the sets A and B defined by
A :=(f0,Φ±(·)
)| f0 ∈ IR, Φ± : I(u0,±, u±) 7→ IR and (28), (31) hold,
where u0,± are given by (19)
B :=(U, V ) | (U, V ) is a bounded self-similar distribution solution of (6), (2)
Remark 3 Note that (28) is equivalent to (12),(13) under additional regularity assumptions
on f . For example, it is sufficient that f be sublinear at least on one side from each point
of IR . The argument is the same that was used in the proof of Proposition 2.
We state the existence and convergence results and the maximum principle for (28) in case
a < b :
Proposition 6 For all continuous function f , all ε > 0 and a, b ∈ IR such that f(u) >
f(a) for all u ∈ (a, b] , there exists a unique Φ that satisfy (28).
1.II.5. Hyperbolic-elliptic case 61
Proof: Using the Maximum Principle below, as in the proof of Proposition 2 we construct a
sequence of solutions Φn to (12) penalized by troncatures. (We need the condition f > f(a)
on (a, b] in order to guarantee Φn > 0 .) As n → +∞ , Φn decrease to a continuous
increasing convex function Φ on [a, b] .
By the Fatou Lemma, Φ ≥ 2εΦf−Φ
in the sense of measures on (a, b) ; this also implies that
Ω := u | Φ(u) = f(u) is of measure 0 , and Φ ≤ f . Besides, if Φ(u) < f(u) , then1
f−Φnand Φn are uniformly bounded in a neighbourhood of u , because Φn are convex
and Φn(b) = Φ(b) < Φ(a) for all n . Therefore we get the equality in (28) outside of
Ω . As in Proposition 2, it follows that Φ is strictly increasing. Other properties in (28)
are now obvious. Thus the existence of solution is shown; the Maximum Principle yields the
uniqueness. ⋄
Lemma 6 (Maximum Principle) Let (a, b) ⊂ IR and Φ,Ψ satisfy, in the sense of (28),
the “inequalities” Φ(u) ≥ G(u,Φ(u), Φ(u)) and Ψ(u) ≥ H(u,Ψ(u), Ψ(u)) , respectively,
with G,H : (a, b)× (−∞,+∞)× (0,+∞) 7→ (0,+∞] .
a) Assume that G(u, z, w) < H(u, ζ, w) for all u such that Φ(u) < Ψ(u) and all z, ζ, w
such that z < ζ . Then Φ ≥ Ψ on [a, b] whenever Φ(a) ≥ Ψ(a) and Φ(b) ≥ Ψ(b) .
b) Assume that G(u, z, w) ≡ H(u, z, w) and increases in z (strictly) and in w ; let
Φ(a) = Ψ(a) or Φ(b) = Ψ(b) . Then (Φ−Ψ) is monotone on [a, b] .
Proof: For instance, let prove a). Take c the point of minimum of Φ − Ψ ; c ∈ (a, b) .
Suppose this minimum is negative. Then Φ(c) < Ψ(c) ≤ f(c) , so that there exists δ1 > 0
such that Φ(·) = G(·,Φ(·), Φ(·)) ∈ L1loc(c, c+ δ1) . Then there exists Φ(c) ; by the choice of
c , we will have Ψ(c+0) ≤ Φ(c) ≤ Ψ(c−0) . The convexity of Ψ implies that Ψ(c) = Φ(c) .
Therefore we have G(u,Φ(u), Φ(u)) < H(u,Ψ(u), Ψ(u)) for u = c , and consequently, for
all u ∈ [c, c + δ2) for some δ2 > 0 . From (28) we have Φ < Ψ in the sense of measures
on [c, c + δ) , where δ := minδ1, δ2 . Therefore Φ − Ψ strictly decreases on [c, c + δ] ,
which contradicts to the choice of c . ⋄Other changes, with respect to the case of monotone f , are minimal. Let only note that
we have to take into account that u0,± are not continuous as functions of f0 , in particular
in the assertion analogous to d) of Lemma 4.
As in the continuous case, we can prove the convergence of Φε± with fixed f0 ; then
establish that (31) is a bijection between f0 and v−−v+ (for instance, under the assumption
f± = ±∞ ); then prove the convergence of Φε± with fixed v− − v+ to the corresponding
convex hulls and apply Lemma 3.
The following theorem holds:
Theorem 2 Suppose lim infu→−∞
f(u) = −∞, lim supu→+∞
f(u) = +∞ . Then for all u±, v± ∈ IR ,
ε > 0 there exists a unique bounded self-similar distribution solution (Uε, V ε) of the problem
(6),(2).
62 Riemann problem for p-Systems
Besides, as ε ↓ 0 (Uε, V ε)(ξ) → (U, V )(ξ) a.e. on IR , where (U, V ) is given by
the formulae (19)-(22), so that (U, V ) is a self-similar distribution solution of the problem
(1),(2).
Remark 4 In this last section we have answered to a question posed by R.James in [Ja80],
the paper from which an extensive study of admissibility for (1),(2) in the non-monotone case
had started. James studies nonlinear elasticity and proposes to define admissible solutions as
limits of the vanishing viscosity approximations of the formUt − Vx = 0
Vt − f(U)x = εVxx,(32)
following the idea defended by Rayleigh ([Ray10]) for the isentropic fluid dynamics. James
calls this criterion the viscoelastic criterion. He proceeds by deducing that the energy should
decrease in time, but this condition is not sufficient for uniqueness for the Riemann problem
(which is well known in the scalar case, in absence of convexity). He proposes then to
obtain more restrictions on admissible shocks either by using the full strength of the viscosity(0
εVxx
), either by replacing this viscosity in (32) by the artificial one used by Dafermos in
[D74],
(εtUxx
εtVxx
)(or by
(0
εtVxx
), which encounters less physical objections).
For the first case, there is a response in the case of flux functions with one inflexion point.
M.Shearer in [Sh82] constructs a Riemann solver which provides a unique solution to (1),(2) for
all data, using classical shocks (always admissible by the viscoelastic criterion) and additional
phase transitions propagating with zero speed. R.Pego in [Pe87] observes that only these two
kinds of shocks can be approximated by travelling waves solutions of (32).
The result of this section give a response for the second case. Evidently, in the Riemann
solver (19)-(22) the only admissible transitions between elliptic and hyperbolic regions or
between two elliptic regions are those with zero speed.
It seems that this last property is not always what one observes in physical systems. The
reason is, dissipation effects captured by vanishing viscosity can coexist with dispersion effects
provoked by capillarity. M.Slemrod in [Sl83] proposes a family of viscosity-capillarity criteria,
regularizing (1) with
(0
εVxx −Aε2Vxxx
). The parameter A ∈ [0, 1/4] regulates phase
transitions; an important feature is that viscosity-capillarity limits admit phase transitions of
non-zero speed whenever A > 0 . Clearly, the self-similar viscosity approach exposed above
is unable to capture this kind of effects. To pursue the study, one has to introduce additional
self-similar dissipation, as it has been done by M.Slemrod and H.Fan in [Sl89, Fan92].
Observations on the difference of diffusive and diffusive-dispersive limits for (1),(2) have
recently gave rise to a theory of non-classical Riemann solvers, in particular for nonconvex scalar
conservation laws. A survey of results on diffusive-dispersive limits and related questions can
be found in [LF98].
CHAPTER 1.III
On Viscous Limit Solutions
to the Riemann Problem
for the Equations of Isentropic Gas Dynamics
in Eulerian Coordinates†
Introduction
In this chapter we study bounded self-similar solutions to the problem
ρt + (ρu)x = 0
(ρu)t + (ρu2 + p(ρ))x = εtuxx(1)
with the initial condition
ρ(0, x) =
ρ+, x > 0
ρ−, x < 0, u(0, x) =
u+, x > 0
u−, x < 0(2)
and establish convergence of solutions as ε ↓ 0 .
Within the framework of isentropic gas dynamics in Eulerian coordinates, (ρ, u) : (t, x) ∈IR+ × IR 7→ (ρ(t, x), u(t, x)) ∈ IR+ × IR corresponds to the density and velocity in gas,
p is the pressure law of the gas, and ε > 0 models small dissipation of the momentum.
We assume that ρ± > 0 and p(·) is continuous strictly increasing on IR+ , normalized by
p(0) = 0 .
Recently, the problem (1),(2) has been treated in [Kim99], following the ideas of [Tz95] (see
also [Tz96]). Under additional assumptions that prohibit vaccuum in the solutions, existence
for the problem (1),(2) was proved. The set of all solutions was shown to be compact in BV ,
and the wave-fan structure of limiting functions as ε ↓ 0 was described.
†This chapter is being prepared upon publication [BA4]
64 Riemann Problem for Gas Dynamics in Eulerian Coordinates
Here we give a description of solutions to (1),(2) that is also valid when a solution contains
a vaccuum state. It also suggests a formula for the limiting function as ε ↓ 0 (cf. Section 3).
Let illustrate the problem with the classical example of γ -laws, i.e. p(ρ) = const · ργ ,γ ≥ 0 . First take γ = 1 . Set f(V ) = −p(1/V ) , k(V ) = 1/V , and consider the problem
Vt − uy = 0
ut − f(V )y = εt(k(V )uy)y(3)
V (0, y) =
1/ρ+, y > 0
1/ρ−, y < 0, u(0, y) =
u+, y > 0
u−, y < 0. (4)
This problem is (1),(2) rewritten in the Lagrangian coordinates (here y is the matherial
coordinate, and V = 1/ρ ), provided ρ > 0 (e.g., see [RoJa], [ChHs]). For the system
(3), the Riemann problem can be studied extensively (cf. Section 4 of Chapter 1.II) by the
method applied in [BA3]. It yields existence and uniqueness of a bounded self-similar solution
for all Riemann data V± > 0, u± ∈ IR , and all ε > 0 . As ε ↓ 0 , the solutions converge
to a function described by an explicit formula, based on use of convex and concave hulls of
the graph of f . In each of this solutions V is bounded, i.e. ρ = 1/V > 0 .Therefore we
can pass to the Eulerian coordinates and deduce the same results for (1),(2). The explicit
formula for the limiting function will use the images of convex/concave hulls of f(V ) under
the transformation
T : [F : V ∈ (0,+∞) 7→ F (V )] 7→ [P : ρ ∈ (0,+∞) 7→ −F (1/ρ)]. (5)
For instance, let [a, b] ⊂ (0,+∞) and F (·) be the concave hull of f(·) on [1/b, 1/a] . Let
P (·) be the function ρ ∈ (0,+∞) 7→ −F (1/ρ) . Then P (·) can be characterized by the
following properties:
(i) P (·) ∈ C[a, b] and P ≤ p on [a, b]
(ii) the function F = T−1P is concave on (1/b, 1/a)
(iii) for all Q(·) that satisfies (i) and (ii), one has P ≥ Q on [a, b].
(6)
In the case p(ρ) = const · ργ with γ > 1 , it is well known that, in general, one cannot
avoid the appearance of vaccuum in solutions of (1),(2) with ε = 0 (a detailed study of
this problem for γ ≥ 1 through construction of wave curves on the half-plane (ρ, u) can be
found in [ChHs]). The same difficulty appears for ε > 0 . This is due to the fact that the
integral
∫ 1
ρ
√d
drp(r)
dr
r=
∫ 1/ρ
1
√d
dvf(v)dv converges as ρ → 0 , which impose a bound
on the size of Riemann data in order to have a bounded solution of (3),(4) (e.g., see Chapter
1.II and [BA3]). We will solve the problem (1),(2) independently (cf. Theorem 1) since we
cannot reduce it to (3),(4) any more†.
†In fact, such reduction is possible, but should involve measure-valued solutions at the limit as ε ↓ 0 ; see
[Wa87] and the footnote in Section 4 of Chapter 1.II.
1.III.0. Introduction 65
Nevertheless, it is still possible to deduce a formula for the limiting function. In fact, (6)
makes sense also for a = 0 (upon formally setting 1/a = +∞ in (ii)), which corresponds
to the presence of a vaccuum state ρ = 0 . More precisely, for all [a, b] ⊂ [0,+∞) there
exists a (unique) function P (·) that satisfies (6). Indeed, for a > 0 it is obvious; for a = 0 ,
we construct P (·) as the decreasing limit of functions Pδ(·) such that Pδ(·)|[0,δ] ≡ p(δ) ,
Pδ(·)|[δ,b] verifies (6) on [δ, b] . Proprieties (i)− (iii) in (6) follow easily from the monotony
and continuity of P (·) . This motivates the following definition.
Definition 1 Let p : IR+ 7→ IR be continuous and strictly increasing. For a ≥ 0 and
b ≥ a , the lower (-1)-hull of p(·) on [a, b] is the function P (·) that verifies (6). For b > 0
and a ≥ b , the upper (-1)-hull of p(·) on [b, a] is the function P (·) that verifies
(i) P (·) ∈ C[b, a] and P ≥ p on [b, a]
(ii) the function F = T−1P is convex on (1/a, 1/b)
(iii) for all Q(·) that satisfies (i) and (ii), one has P ≤ Q on [b, a].
(7)
For a′, b′ ∈ (IR+)2 denote the segment [mina′, b′,maxa′, b′] by I(a′, b′) . The (-1)-hull
of p(·) on I(a′, b′) is the lower (-1)-hull on [a′, b′] in case a′ ≤ b′ and the upper (-1)-hull
on [b′, a′] in case a′ > b′ .
In Section 3 we consider the system
ρt + (ρu)x = 0
(ρu)t + (ρu2 + p(ρ))x = 0(8)
and, using (-1)-hulls of p(·) , give a formula for the unique solution of the Riemann problem
(8),(2) that can be obtained as a limit of self-similar bounded weak solutions of (1),(2) with
ε = εn for a sequence εn ↓ 0 .
The approach by self-similar viscosity limits has been used in [Ka59, Tu64, Tu66, Tu73,
D73a, D74, DDp76] in the context of admissibility of weak solutions to the Riemann problem
for hyperbolic systems of conservation laws. In [D89], Dafermos postulated it as the wave fan
admissibility criterion. It has been successfully tested on various special systems and viscosity
matrices; see [Tz98], [Tz96] for a survey of recent results in this direction. In particular, an
analysis of the problem (8),(2) regularized with the self-similar viscosity εt
(ρ
ρu
)is carried
out in [SlTz89], covering among others the cases where vaccuum is present. Within the same
framework, a special attention to the formation of the vaccuum state is paid in [Fan91]. In
[Tz96] the existence of an admissible solution is proved for a large class of strictly hyperbolic
systems with close Riemann data, using the identity self-similar viscosity matrix.
66 Riemann Problem for Gas Dynamics in Eulerian Coordinates
The main result of this chapter is that, for the case of the degenerate viscosity matrix(0 0
0 1
)and under the additional assumption
(a) either p(ρ) → +∞ as ρ→ +∞,
(b) or p ∈ W 1,1loc (R,+∞) for some R > 0, and d
∫ +∞
R
√d
drp(r)
dr
r= +∞,
(9)
there is global existence and uniqueness of an admissible weak solution to the Riemann problem
for the nonstrictly hyperbolic system (8) in the sense of the wave fan admissibility criterion
(cf. Theorem 2). We describe the structure of this solution in Section 3.
1 Some useful properties of viscous approximations
Let fix ε > 0 . We are concerned with bounded self-similar distribution solutions to the
problem (1),(2).
Definition 2 A pair of functions (ρ, u) : IR+ × IR 7→ IR+ × IR is a solution of the problem
(1),(2) if for all k > 0 , (ρ, u)(t, x) = (ρ, u)(kt, kx) for a.a. (t, x) ∈ IR+ × IR , ρ, u ∈L∞(IR+ × IR) , (1) is fulfilled in D′(IR+ × IR) , and
ess limt↓0
(‖ρ(t, ·)− ρ(0, ·)‖L1(−R,R) + ‖u(t, ·)− u(0, ·)‖L1(−R,R)
)= 0 (10)
for all R > 0 , where ρ(0, ·), u(0, ·) are given by (2).
We will denote x/t by ξ and ambiguously use the same notation for a self-similar function
of the variables (t, x) and the corresponding function of ξ .
Lemma 1 A pair (ρ, u) is a solution of (1),(2) in the sense of Definition 2 if and only if the
following conditions are fulfilled:
(i) there exist continuous bounded functions ρ, u : IR 7→ IR , with u′(·) and (·−u(·))ρ′(·)continuous, such that (ρ, u)(t, x) = (ρ, u)(x/t) for a.a. (t, x) ∈ (0,+∞)× IR ;
(ii) there exists a constant C ∈ IR such that one has
εu′(ξ) = −∫ ξ
0
(ζ − u(ζ))2ρ′(ζ) dζ + f(ρ(ξ)) + C, (11)
ρ(ξ)u′(ξ) = (ξ − u(ξ))ρ′(ξ), (12)
limξ→±∞
ρ(ξ) = ρ±, limξ→±∞
u(ξ) = u±. (13)
1.III.1. Useful properties of viscous approximations 67
Besides, ther exists a unique ξ0 such that u(ξ0) = ξ0 . In case ρ(ξ0) > 0 there exist
ξ± ∈ IR , ξ− ≤ ξ0 ≤ ξ+ , such that both ρ(·) and u(·) are constant on (ξ−, ξ+) and
strictly monotone on (−∞, ξ−) and (ξ+,+∞) .
In case ρ(ξ0) = 0 , ξ0 is the unique vaccuum point in the solution, ξ± = ξ0 , and the same
monotony properties hold.
Moreover, u′(ξ) 6= 0 for all ξ ∈ (−∞, ξ−) ∪ (ξ+,+∞) , u′(ξ0) = 0 in case ρ(ξ0) > 0 and
0 ≤ u′(ξ0) < 1 in case ρ(ξ0) = 0 .
Remark 1 We see from Lemma 1 that in case ε > 0 , solutions of (1),(2) contain at most
one vaccuum point.
Proof: The proof consists of four steps.
I) Let (ρ, u) be solution of (1),(2). Then (ρ, u)(t, x) = (ρ, u)(x/t) and we have in D′(IR)
−ξρ′ + (ρu)′ = 0
−ξ(ρu)′ + (ρu2 + p(ρ))′ = εu′′.(14)
It follows that εu′′ = −ξ2ρ′ + (ρu2 + p(ρ))′ = −(ξ2ρ)′ + 2ξρ + (ρu2 + p(ρ))′ in D′(IR) ,
whence u′ ∈ L∞loc(IR) and u ∈ C(IR) . Thus ρu′, (ρu)u′ are well defined in D′(IR) , hence
ρ′u , ρ′u2 as well. Therefore we obtain
−(ξ − u)ρ′ + ρu′ = 0 in D′(IR) (15)
and
−(ξ − u)2ρ′ + p(ρ)′ = εu′′ in D′(IR). (16)
It follows that −(ξ − u)ρ =
∫ ξ
0
ρ(ζ) dζ + const ∈ C(IR) . Consequently, −(ξ − u)2ρ +
p(ρ)− εu′ =
∫ ξ
0
2(ζ − u(ζ))ρ(ζ) dζ + const ∈ C1(IR) . Thus
p(ρ(·))− εu′(·) ∈ C(IR). (17)
II) Now consider the function ξ 7→ ξ−u(ξ) . It is continuous and tends to ±∞ as ξ → ±∞ .
Therefore there exist finite η− := minξ | u(ξ) = ξ and η+ := maxξ | u(ξ) = ξ . One
has ξ − u(ξ) < 0 on (−∞, η−) and ξ − u(ξ) > 0 on (η+,+∞) . By (15), on each of
these intervals one has ρ′ ∈ L∞loc . Therefore ρ is continuous on these intervals; by (17), so
does u′ ; by (15), so does ρ′ .
Let prove that ρ, u are monotone on (−∞, η−) and (η+,+∞) . Take ξ+ := supξ ≥η+ |ρ|(η+,ξ) ≡ const . First show that u is strictly monotone on (ξ+,+∞) . Indeed, suppose
the contrary. Then there exists c ∈ (ξ+,+∞) and 0 < δ < c − ξ+ such that u′(c) = 0
68 Riemann Problem for Gas Dynamics in Eulerian Coordinates
and u′ 6= 0 on (c− δ, c) . From (15) we have ρ′ = ρu′
ξ−u≥ 0 on (c− δ, c) , so that p(ρ(·))
is non-decreasing on [c− δ, c] . Hence by (15),(16) we have
εu′(ξ) =
∫ c
ξ
(ζ − u)ρu′ dζ + p(ρ(ξ))− p(ρ(c)) ≤∫ c
ξ
(ζ − u)ρu′ dζ
pointwise on [c − δ, c] . Choosing a sequence ξn ↑ c such that u′(ξn) = maxξ∈[ξn,c] u′(ξ) ,
we get εu′(ξn) ≤ u′(ξn) O(c − ξn) , where O(·) is the Landau symbol. As n → ∞ , we
deduce ε ≤ 0 , which is a contradiction.
We conclude that u is strictly monotone on (ξ+,+∞) ; hence ρ is monotone on
(η+,+∞) . Similarly, there exists ξ− ≤ η− such that u, ρ are strictly monotone on
(−∞, ξ−) and ρ ≡ const on [ξ−, η−) .
III) Let investigate the behaviour of ρ, u on the interval [η−, η+] . First note that there exist
finite limits ρ(η± ± 0) ; by (17), there exist finite limits u′(η± ± 0) . We notice that, first,
u′(η+ + 0) = 0 in case ρ(η+ + 0) > 0 , and 0 ≤ u′(η+ + 0) < 1 in case ρ(η+ + 0) = 0 .
The same relation exists between u′(η− − 0) and ρ(η− − 0) .
Indeed, let ρ(η++0) > 0 . By (15), we have
∫ η++1
η+
(ln ρ(ζ)
)′dζ =
∫ η++1
η+
u′(ζ)
ζ − u(ζ)dζ .
The integral on the left converge, therefore the limit u′(η + 0) is necessarily 0 . Further, let
ρ(η++0) = 0 . Then ρ is non-decreasing on (η+,+∞) , hence u′(η++0) ≥ 0 by (15). On
the other hand, the definition of η+ trivially implies that u′(η+ + 0) ≤ 1 . Besides, suppose
u′(η+ + 0) = 1 . From (16),(15) we obtain
ε(1− u′(ξ)) =
∫ ξ
η+
(ζ − u(ζ))ρ(ζ)u′(ζ) dζ − p(ρ(ξ))
for all ξ ∈ (η+,+∞) . Since ρu′ is continuous on this interval and tends to 0 as ξ ↓ η+ ,
there exists δ > 0 such that
ε(ξ − u(ξ))′ ≤∫ ξ
η+
(ζ − u(ζ)) dζ (18)
for all ξ ∈ (η+, η+ + δ) . Setting g(ξ) := ξ − u(ξ) and h(ξ) :=√εg(ξ) +
∫ ξ
η+
g(ζ)dζ , we
obtain from (18) that h ∈ C1[η−,+∞) and h′(ξ) ≤ 1/√ε h(ξ) . Since h(η+) = 0 , we get
h ≡ 0 on [η+, η+ + δ] by the Gronwall inequality. This contradicts the definition of η+ .
Thus, definitively, 0 ≤ u′(η+ + 0) < 1 . The proof for η− − 0 in the place of η+ + 0 is
likewise.
Now we analyse separately the three cases:
a) η− = η+ =: ξ0 and one of the limits ρ(ξ0 ± 0) is non-zero. Then the two limits
coincide. Indeed, let ρ(ξ0 + 0) > 0 . First assume ρ(ξ0 − 0) > 0 ; in this case u′(ξ0 ± 0) =
0 , so that ρ(ξ0 ± 0) coincide by (17). Further, assume ρ(ξ0 − 0) = 0 . In this case
1.III.1. Useful properties of viscous approximations 69
p(ρ(ξ0+0))−εu′(ξ0+0) = p(ρ(ξ0+0)) > 0 , and p(ρ(ξ0−0))−εu′(ξ0−0) = −εu′(ξ0−0) ≤ 0 ,
which contradicts (17). Hence we see that ρ, u′ are both continuous on IR . Therefore
(15),(16) can be rewritten under the form (11),(12).
We see that in this case there exists a unique ξ0 such that u(ξ0) = ξ0 , and we have
ρ(ξ0) > 0 and u′(ξ0) = 0 . Besides, with ξ± defined above, we find that ρ, u are constant
on [ξ−, ξ+] and that u′ , consequently ρ′ , are different from 0 on (−∞, ξ−)∪ (ξ+,+∞) .
b) η− = η+ =: ξ0 and ρ(ξ0 ± 0) = 0 . Then ρ, u′ are continuous on IR , and (11),(12)
follow. Further, ξ = ξ0 is the unique vaccum point. Indeed, for instance, take ξ+ =
supξ | ρ(ξ) = 0 . We have
∫ ξ++1
ξ+
(ln ρ(ζ)
)′dζ =
∫ ξ++1
ξ+
u′(ζ)
ζ − u(ζ)dζ , but this time the
left-hand side integral diverge. It follows that ζ − u(ζ) → 0 as ξ ↓ ξ+ , whence ξ+ = ξ0 .
We see that in this case there exists a unique ξ0 such that u(ξ0) = ξ0 , and we have
ρ(ξ0) = 0 and 0 ≤ u′(ξ0) < 1 ; besides, u′ , ρ′ are different from 0 on IR \ ξ0 .
c) η− < η+ Actually, we will show that this case is impossible. Indeed, from (15) we have
ρ = ((ξ − u)ρ)′ in D′(IR) . Therefore
∫ η+
η−
ρ(ζ) dζ =[(ξ − u(ξ))ρ(ξ)
]∣∣∣η+
η−= 0 , so that
ρ|(η−,η+) ≡ 0 . Hence u′|(η−,η+) ≡ const by (16); taking into account that u(η±) = η± , we
find u′(η± ∓ 0) = 1 . Besides, 0 ≤ u′(η± ± 0) < 1 in all cases; arguing as in case a), we get
a contradiction with (17).
We conclude that ρ(·), u(·) satisfy (11),(12) and have all continuity and structure prop-
erties in Lemma 1. The monotony of ρ, u trivially implies that (13) is satisfied if and only if
self-similar ρ(·, ·), u(·, ·) satisfy (10).
IV) Conversely, (11)-(13) together with the continuity of ρ(·), u(·), u′(·) and (· − u(·))ρ′(·)imply that (ρ, u)(t, x) := (ρ, u)(x/t) is a solution of (1),(2). Indeed, (14) is straightforward.
Besides, (11),(12) yield the monotony of ρ(·), u(·) at ±∞ . Thus (10) holds also. ⋄
Using the results of Lemma 1, set ρ0 := ρ(ξ0) and k := u′(ξ0) ; define σ := ρ0 − k .
Note that σ ∈ (−1,+∞) and
ρ0 = (σ)+ = maxσ, 0, k = (σ)− = max−σ, 0. (19)
Further, split IR into the three (may be empty) intervals (−∞, ξ−) , (ξ−, ξ+) , (ξ+,∞) . Let
us inverse ρ(·) on (−∞, ξ−) and (ξ+,+∞) . We will ambiguously use the same notation
for the function ρ(·) and the independent variable ρ ∈ IR+ . The functions
ρ−1− : I(ρ0, ρ−) 7→ (−∞, ξ−), ρ−1
+ : I(ρ0, ρ+) 7→ (ξ+,+∞)
are well defined. Set
Πε±(ρ; σ) :=
∫ ρ
(σ)+
[ρ−1± (r)− u(ρ−1
± (r))]2dr − C (20)
70 Riemann Problem for Gas Dynamics in Eulerian Coordinates
for ρ ∈ I(ρ0, ρ±) , where C is taken from (11). Let use the simplified notation Π±(·) for
Πε±(·; σ) whenever ε, σ are fixed; besides, let ˙ denote derivation with respect to ρ . We
can rewrite (11) under the form
εu′(ξ) = p(ρ(ξ))− Π±(ρ(ξ)). (21)
Since 0 /∈ I(ρ0, ρ±) and u′(ξ) is shown to be non-zero on (−∞, ξ−)∪(ξ+,+∞) , we deduce
that Π± ∈ C2(I(ρ0, ρ±)) and that one has
ρΠ± + 2Π± =2εΠ±
f − Π±, Π± > 0 and sign(p−Π±) = sign(ρ± − ρ0) (22)
on I(ρ0, ρ±) . Further, Π± can be extended to I(ρ0, ρ±) by continuity. By (21) and
Lemma 1 Π±(·) admits finite limits at ρ0 , and we can assign
Π±(ρ0) = p(ρ0)− εk, (23)
where ρ0, k are defined by (19). Besides,
Π±(ρ±) = p(ρ±). (24)
Indeed, the right-hand side in (21) admits finite limits as ρ(ξ) → ρ± , because, in case
ρ0 < ρ± , Π±(·) are increasing and bounded from above and, in case ρ0 > ρ± , Π±(·) are
concave and bounded from below. We deduce u′(ξ) → 0 as ξ → ±∞ , since u has finite
limits at ±∞ .
In addition, in order to cover the case of ξ− = −∞ (i.e., ρ0 = ρ− ) and/or ξ+ = +∞(i.e., ρ0 = ρ+ ), we just define Π−(·) and/or Π+(·) by (24).
Finally, (20) and (12) yield u′(ξ) = ±√
Π±(ρ)ρ′(ξ)
ρfor all ξ ∈ (−∞, ξ−) and all
ξ ∈ (ξ+,+∞) , respectively. Since u(ξ−) = u(ξ+) , (13) yields that
u+ − u− =
∫ ρ+
ρ0
√Π+(r)
dr
r+
∫ ρ−
ρ0
√Π−(r)
dr
r(25)
and the integrals in the right-hand side of (25) are finite.
We have established the following result.
Proposition 1 Let (ρ, u) be a solution of (1),(2) in the sense of Definition 2. Then there
exist σ ∈ (−1,+∞) , ρ0 ∈ [0,+∞] , k ∈ [0, 1) , and functions Π± ∈ C(I(ρ0, ρ±)) ∩C2(I(ρ0, ρ±)) such that (19) and (22)-(25) hold.
The converse assertion is also true.
1.III.1. Useful properties of viscous approximations 71
Proposition 2 Let σ ∈ (−1,+∞) and ρ0, k be defined by (19). Let Π± ∈ C(I(ρ0, ρ±))∩C2(I(ρ0, ρ±)) satisfy (22)-(25). Then there exists a solution (ρ, u) of (1),(2) in the sense
of Definition 2, and it is given by
ρ(t, x) = ρ(x/t) =
[Ξε
−]−1(x/t), x/t < ξ+
[Ξε+]
−1(x/t), ξ− < x/t≡
[Ξε−]
−1(x/t), x/t < ξ−
ρ0, ξ− < x/t < ξ+
[Ξε+]
−1(x/t), ξ+ < x/t,
(26)
u(t, x) = u(x/t) =
Uε− [Ξε
−]−1(x/t), x/t < ξ−
Uε−(ρ0) = Uε
+(ρ0), ξ− < x/t < ξ+
Uε+ [Ξε
+]−1(x/t), ξ+ < x/t,
(27)
where
Uε±(ρ) := u± ∓
∫ ρ±
ρ
√Π±(r)
dr
rfor ρ ∈ I(ρ0, ρ±), (28)
Ξε±(ρ) := Uε
±(ρ)±√Π±(ρ) for ρ ∈ I(ρ0, ρ±), (29)
and ξ± are defined by
ξ± := limρ∈I(ρ0,ρ±), ρ→ρ0
Ξε±(ρ) if ρ0 6= ρ±;
ξ− := −∞ and/or ξ+ := +∞ if ρ0 = ρ− and/or ρ0 = ρ+.
(30)
Proof: The cases ρ0 = ρ+ , ρ0 = ρ− are trivial; assume ρ0 6= ρ± . Let Uε±(·) , Ξε
±(·)be defined by (28),(29). Note that both integrals in (28) necessarily converge as ρ → ρ0 ,
ρ ∈ I(ρ0, ρ±) . Indeed, in case ρ0 = 0 both are positive and thus finite, by (25). In case
ρ0 > 0 , Π± are bounded as ρ → ρ0 , by monotony of ρ2Π± which is evident from (22).
Therefore Uε±(ρ0) are well defined; by (25) they coincide. Note also that
Ξε±(ρ) = ±ρΠ±(ρ) + 2Π±(ρ)
2ρ√
Π±(ρ)= ±
ε√Π±(ρ)
ρ(p(ρ)−Π±(ρ))(31)
are continuous and non-zero on I(ρ0, ρ±) , so that ξ± and [Ξε±]
−1 are well defined in the
graph sense. We see from (25),(28),(29) that
ξ+ − ξ− = limρ∈I(ρ0,ρ−), ρ→ρ0
√Π−(ρ) + lim
ρ∈I(ρ0,ρ+), ρ→ρ0
√Π+(ρ) ≥ 0. (32)
Clearly, ρ, u ∈ C(IR)∩C1(IR\ξ−, ξ+) ; besides, (26)-(29),(31) and (24) yield (13),(21)
for all ξ 6= ξ± . In fact, we could show that (24),(22) imply that√
Π±(ρ) → +∞ as
ρ → ρ± , ρ ∈ I(ρ0, ρ±) , so that Ξε± map I(ρ0, ρ±) onto (ξ+,+∞) and (−∞, ξ−) ,
72 Riemann Problem for Gas Dynamics in Eulerian Coordinates
respectively. Still (24) yields (21) for ξ outside the range of Ξε± even if we admit that Ξε
±
can be bounded. Futhermore, by (31),(21),(29) and (26),(27) we have for all ξ 6= ξ±
(ξ − u(ξ))ρ′(ξ) = ±ρ(p(ρ(ξ))−Π±(ρ(ξ)))
±ε = ρ(ξ)u′(ξ); (33)
whence (12) follows for ξ 6= ξ± .
Now consider the two possibilities:
a) ρ0 > 0 . Then u′(ξ± ± 0) = 0 by (21) and (24), while in case ξ− < ξ+ we have
u′(ξ± ∓ 0) = 0 by (27). Thus u ∈ C1(IR) and, by (33), (· − u(·))ρ′(·) is continuous on
IR . It now follows that (11),(12) hold everywhere; besides, (13) is obvious.
b) ρ0 = 0 . Then Π±(ρ) → 0 as ρ → ρ0 , ρ ∈ I(ρ0, ρ±) . Indeed, by (22) we have for
ρ ∈ I(ρ0, ρ±)
∫ (ρ0+ρ+)/2
ρ
(ln Π±(r)
)·dr =
∫ (ρ0+ρ+)/2
ρ
2
r
( ε
p(r)−Π±(r)− 1)dr.
By (23),(19) it follows that ε/(p(r)− Π±(r)) − 1 → 1/k − 1 > 0 as ρ → ρ0 . Therefore
the last integral diverge, hence ln Π±(ρ) → −∞ as ρ → ρ0 , ρ ∈ I(ρ0, ρ±) . We conclude
by (32) that ξ− = ξ+ ; by (21) and (23), u′(ξ± ± 0) = k . Thus again u′(·) , (· − u(·))ρ′(·)are continuous on IR , and (11)-(13) hold.
By Lemma 1, it follows that (ρ, u) is a solution of (1),(2). ⋄
2 Existence and uniqueness of viscous approximations
In Section 1, we have reduced the problem (1),(2) to finding σ ∈ (−1,+∞) , ρ0 , k and a
pair of functions Π± ∈ C(I(ρ0, ρ±))∩C2(I(ρ0, ρ±)) that satisfy (19) and (22)-(25). In this
section we prove that such σ, ρ0, k,Π±(·) do exist (and are unique), provided p(·) satisfies
(9) (cf. Proposition 3 and Lemmae 4,5,6 below). We also prove two preliminary convergence
results (cf. Lemma 3 and Proposition 4).
Start by fixing σ ∈ [−1,+∞) and b ∈ (0,+∞) . Set a := (σ)+ , k := (σ)− and
consider the problem of finding Π ∈ C(I(a, b) ∩ C2(I(a, b)) satisfying
ρΠ(ρ) + 2Π(ρ) =2εΠ(ρ)
p(ρ)−Π(ρ)with Π(ρ) > 0 and (b− a)(p(ρ)− Π(ρ)) > 0
for all ρ ∈ I(a, b),
Π(a) = p(a)− εk,
Π(b) = p(b).
(34)
Proposition 3 There exists a unique solution to the problem (34).
Let Πε(·) denote the solution of (34) corresponding to ε , ε > 0 . Recall Definition 1.
1.III.2. Existence and uniqueness of viscous approximations 73
Proposition 4 As ε ↓ 0 , Πε converge to the (-1)-hull of p(·) uniformly on I(a, b) .
First note that the equation in (34) is still in the scope of the maximum principle (cf.
Chapter 1.II). For the sake of completeness, we restate it here for the case a < b .
Lemma 2 [Maximum Principle] Let Π,Υ ∈ C[a, b]∩C2(a, b) and satisfy, for all ρ ∈ (a, b) ,
the equations Π(ρ) = G(ρ,Π(ρ), Π(ρ)) and Υ(ρ) = H(ρ,Υ(ρ), Υ(ρ)) , respectively, with
some G,H : (a, b)× IR× (0,+∞) 7→ (0,+∞] .
a) Assume that G(ρ, z, w) < H(ρ, ζ, w) for all ρ ∈ (a, b) such that Π(ρ) < Υ(ρ)
and all z, ζ, w such that z < ζ . Then Π ≥ Υ on [a, b] whenever Π(a) ≥ Υ(a) and
Π(b) ≥ Υ(b) .
b) Assume that G(ρ, z, w) ≡ H(ρ, z, w) , increases in w and strictly increases in z ; let
Π(a) = Υ(a) or Π(b) = Υ(b) . Then (Π−Υ) is monotone on [a, b] .
Proof: The proof is straightforward. ⋄
Secondly, note the following lemma, which will also be an ingredient of the convergence
proof in Section 3. Recall (5); as in Definition 1, we understand 1/a as +∞ in case a = 0 .
Lemma 3 Let [a, b] ⊂ IR+ and P εε≥0 ⊂ C[a, b] be a set of functions such that F ε =
T−1P ε are concave on (1/b, 1/a) . Assume that, for all ρ ∈ [a, b] , P ε(ρ) converge to
P 0(ρ) as ε ↓ 0 . Then the following assertions hold.
(a) This convergence is uniform on each segment [c, d] ⊂ (a, b) , and for each ε , P ε
is absolutely continuous on all segment [c, d] ⊂ (a, b) . Moreover, P ε are bounded
uniformly in ε a.e. on all segment [c, d] ⊂ [a, b) such that c > 0 .
(b) For all sequence εn ↓ 0 , P εn → P 0 a.e. on (a, b) ; the convergence take place
everywhere on (a, b) in case P εn, P 0 ∈ C1(a, b) .
(c) Let P ε be increasing and
Ξε±(ρ) := const∓
∫ b
ρ
√P ε(r)
dr
r±√P ε(ρ). (35)
Then Ξε±(·) are a.e. defined monotone functions on (a, b) , so that [Ξε
±]−1 are well
defined in the graph sense.
(d) For P ε increasing, one has
∫ b
a
√P εn(r)
dr
r→∫ b
a
√P 0(r)
dr
rfor all sequence εn ↓
0 , uniformly in ρ ∈ [c, d] for all [c, d] ⊂ [a, b] such that c > 0 .
(e) With the notation of (c), for all sequence εn ↓ 0 one has [Ξεn+ ]−1(ξ) → [Ξ0
+]−1(ξ) for
all ξ such that [Ξ0+]
−1 is continuous at the point ξ . The same holds with [Ξεn− ]−1 ,
[Ξ0−]
−1 in the place of [Ξεn+ ]−1 , [Ξ0
+]−1 .
74 Riemann Problem for Gas Dynamics in Eulerian Coordinates
Analogous properties hold if P ε are defined on [b, a] ⊂ IR+ \ 0 and F ε = T−1P ε are
convex on (1/a, 1/b) .
Proof: Since F ε = T−1P ε are concave, they are differentiable a.e. Moreover, the conver-
gence of P ε(ρ) to P 0(ρ) implies that F ε(1/ρ) → F ε(1/ρ) as ε ↓ 0 . Therefore F ε → F 0
uniformly on all segment [1/d, 1/c] ⊂ (1/b, 1/a) , and d/dV F εn → d/dV F 0 a.e. on
(1/b, 1/a) as ε ↓ 0 . Since P ε(ρ) = 1/ρ2 d/dV F ε(V ) whenever d/dV F ε(V ) exists, (a)
and (b) are evident.
Further, d/dV F ε ≥ 0 in case (c). Substituting v = 1/r in the integral in (35), we
obtain
Ξε±(ρ) = const±
∫ 1/ρ
1/b
(√d
dVF ε(1/ρ)−
√d
dVF ε(v)
)dv ± 1/b
√d
dVF ε(1/ρ),
which is monotone because d/dV F ε is monotone; hence (c). Besides, (d) follows from the
continuity and convergence of F ε(·) at V = 1/b . Indeed, one has for ρ > c > 0
∣∣∣∫ b
ρ
√P εn(r)
dr
r−∫ b
ρ
√P 0(r)
dr
r
∣∣∣ ≤∫ 1/ρ
1/b
∣∣∣√
d
dVF εn(v)−
√d
dVF 0(v)
∣∣∣ dv. (36)
Take δ > 0 and integrate |√d/dV F εn(V )−
√d/dV F 0(V ) | separately over (1/b, 1/b+δ)
and (1/b + δ, 1/c) . For all δ > 0 , the second integral vanishes as εn ↓ 0 , by (a) and the
Lebesgue theorem. Besides, the first one can be made as small as desired by choosing δ small
enough, because one has
∫ 1/b
1/b+δ
√d
dVF εn(v) dv ≤
∫ 1/b
1/b+δ
(1 +d
dVF εn(v)) dv = δ + (F εn(1/b+ δ)− F εn(1/b)) ≤
≤ 2δ + (F 0(1/b+ δ)− F 0(1/b))
for εn sufficiently small. Hence the left-hand side of (36) can be made as small as desired
uniformly in ρ ∈ [c, d] . Moreover, if a > 0 , we can take c = a in the reasoning above and
prove (d).
Finally, (b) and (d) imply that Ξεn± → Ξ0
± a.e. on (a, b) . It is classical in the basic
probability theory that the a.e. convergence of monotone functions (interpreted as random
variables) implies the pointwise convergence of their graph inverse functions (interpreted as
their distribution functions) at the points of continuity of the limit (e.g., cf. [Sv]). Thus (e)
follows.
The case of convex F ε is similar. ⋄
Proof of Proposition 3: The case a = b is trivial; for definiteness, assume a < b ,
which is the most complex case. The uniqueness is immediate from Lemma 2. Besides, for
all a > 0 we could prove the existence directly by penalisation of the right-hand side of (34),
applying Lemmae 2,4 and following the corresponding proof in Chapter 1.II. Instead, we will
1.III.2. Existence and uniqueness of viscous approximations 75
perform the transformation T−1 : [ρ ∈ (a, b) 7→ Π(ρ)] 7→ [V ∈ (1/b, 1/a) 7→ −Π(1/V )] . It
reduces the equation in (34) to the equation
d2
dV 2Φ(V ) =
1
V
2ε d/dV Φ(V )
f(V )− Φ(V )(37)
with d/dV Φ > 0 and Φ > f on (1/b, 1/a) , where Φ = T−1Π and f = T−1p . This
equation only differs from the one that appears in Chapter 1.II by the factor 1/V in the
right-hand side. This factor is continuous and bounded on (1/b, 1/a) since a > 0 , so
that the proof in Chapter 1.II applies without any further modification. Hence there exists
a strictly increasing concave solution Φ ∈ C[1/b, 1/a] ∩ C2(1/b, 1/a) to the problem (37),
Φ(1/a) = f(1/a) , Φ(1/b) = f(1/b) . Therefore Π = TΦ ∈ C[a, b] ∩ C2(a, b) and Π
verifies (34).
For a = 0 , let first find, for all δ ∈ (0, b) a function Πδ ∈ C[0, b] ∩ C2(δ, b) such that
ρΠδ + 2Πδ =2εΠδ
p− Πδ
with Πδ > 0, p−Πδ > 0 on [δ, b);
Πδ(b) = p(b), Πδ|[0,δ] = −kε.
The proof of existence goes on as above. By Lemma 2(b), there exists a function Π on
[0, b] such that Πδ ↑ Π as δ ↓ 0 . Furthermore, applying one more time the same proof
from Chapter 1.II, this time to the functions T−1Πδ , we infer that T−1Π ∈ C2(1/d, 1/c)
and T−1Π satisfies (37) on (1/d, 1/c) for all segment [1/d, 1/c] ⊂ (1/b,+∞) . Thus
Π ∈ C2(0, b) , and the equation in (34) holds. Besides, the continuity of Π at 0 and b
follows from Lemma 2(a) by comparison with special solutions of the equation in (34), as in
Chapter 1.II. Thus Π(0) = −kε and Π(b) = p(b) by the construction of Πδ . ⋄
Proof of Proposition 4: Let us adapt the proof from Chapters 1.I,1.II. Let a < b ,
and P (·) be the (-1)-hull of p(·) on [a, b] . Take α > 0 and construct a barrier function
Υα ∈ C2(a, b) ∩ C[a, b] such that ρΥα + 2Υα ≥ m(α) > 0 and α/2 ≤ P − Υα < α
on (a, b) . Then apply Lemma 2(a) to Πε and Υα . By Lemma 3(a) 1/ρ Πε is uniformly
bounded on all segment [c, d] ⊂ (a, b) . It follows that P ≥ Πε ≥ Υα on [a, b] for all ε
sufficiently small. ⋄
Finally, note the following property of solutions of (34).
Lemma 4 Let Π(·; σ) verify (34) with a = 0 , k = −σ ∈ [0, 1) . Then Π(ρ; σ) → 0 as
ρ → 0 , and the integral S(σ) =
∫ b
0
√Π(r; σ)
dr
ris finite. In addition, S(σ) ↑ +∞ as
σ ↓ −1 .
Proof: Clearly, there is no problem in convergence of the integral on the upper limit. Let
prove that S(σ) converge at the lower limit. Set κ = 1− (1− k)/2 ∈ [1/2, 1) . Since Π, p
are continuous on [0, b] and p(0)−Π(0; σ) = kε , there exists δ > 0 such that p−Π ≤ κε
76 Riemann Problem for Gas Dynamics in Eulerian Coordinates
on (0, δ) . By (34), one has on (0, δ) Π(δ; σ) ≥ Π(ρ; σ)+
∫ δ
ρ
( εκε
− 1)2rΠ(r; σ) dr .
Hence Π(ρ; σ) ≥ const ρ−2(1/κ−1) by the Gronwall inequality, so that the first two assertions
are evident.
Now consider the function Π(·;−1) . If S(−1) diverge, then limσ↓−1 S(σ) = +∞ ,
by Lemma 2(b) and the Levi theorem. Assume that S(−1) < +∞ ; we will arrive to a
contradiction. Indeed, there are two cases.
a) lim supρ↓0 Π(ρ; σ) = 0 . Proceeding as in Proposition 2, we can therefore define ξ+ :=
limρ↓0
(∫ ξ+
ρ
√Π+(r;−1)
dr
r+
√Π+(ρ;−1)
)= S(−1) . Introduce the C1 functions r, u :
ξ ∈ [ξ+,+∞) 7→ ρ(ξ), u(ξ) by formulae (26),(27) and rewrite (34) for ξ, ρ, u . This yields
ρ(ξ+) = 0 , ξ+ − u(ξ+) = 0 , u′(ξ+ + 0) = 1 and
ε(u′(ξ)− u′(ξ+ + 0)) = −∫ ξ
ξ+
(ζ − u(ζ))ρ(ζ)u′(ζ) dζ + p(ρ(ξ)).
for all ξ > ξ+ . As it is shown in part III of the proof of Lemma 1, these properties are
incompartible.
b) lim supρ↓0 Π(ρ; σ) = l2 > 0 . Take κ ∈ (0, 1) . There exists δ0 ∈ (0, b) and σ0 ∈(−1/2,−1) such that p(ρ) − Π(ρ; σ0) ≥ κε for all ρ ∈ [0, δ0] . Take δ1 ∈ (0, δ0] such
that Π(δ1;−1) ≥ l2/2 . Note that, by Lemma 2(a), Π(·; σ) ↓ Π(·;−1) as σ ↓ −1 .
By Lemma 3(b), we also have Π(δ1; σ) → Π(δ1;−1) as σ ↓ −1 . Therefore there exists
σ1 ∈ [σ0,−1) such that Π(δ1; σ1) ≥ l2/4 ; in addition, p(ρ) − Π(ρ; σ1) ≥ κε for all
ρ ∈ [0; δ1] . Now (34) yields Π(δ1; σ1) ≤ Π(ρ; σ1) +
∫ δ1
ρ
( εκε
− 1)2rΠ(r; σ1) dr for all ρ ∈
[0, δ1] . Applying the Gronwall inequality and performing easy calculations, we get S(σ1) ≥∫ δ1
0
√Π(r; σ1)
dr
r≥ l
2δ1/κ−1 δ
1−1/κ
1/κ− 1=
l
2
κ
1− κ. Letting κ ↓ 1 , we conclude that S(-
1) = limσ↓−1 S(σ) = +∞ , which contradicts our assumption. ⋄
For σ ∈ (−1,+∞) denote by Sε±(σ) the integrals
∫ ρ±
(σ)+
√Πε
±(r; σ)dr
r, where Πε
±(·; σ)denotes the unique solution of (22)-(24), according to Proposition 3. By Lemma 4, Sε
±(σ)
are finite for σ ∈ (−1, 0] ; evidently, it is also true in case σ ∈ (0,+∞) .
The next step consists in varying σ in order to satisfy the condition (25), i.e.,
u+ − u− = Sε+(σ) + Sε
−(σ),
which is now shown to be meaningful. We will prove that, for ρ± and ε fixed, (25) establishes
a bijection between σ ∈ (−1,+∞) and u+ − u− ∈ IR , provided p(·) satisfies (9). For
ρ0 = (σ)+ ∈ [0,+∞) , denote by P±(·; ρ0) the (-1)-hulls of p(·) on I(ρ0, ρ±) , respectively.
It will be convenient to extend Πε±(·; σ) , P±(·; ρ0) to continuous functions on IR+ , by
setting them constant on each component of IR+ \ I(ρ0, ρ±) . As in Chapter 1.II, we have
the following two results.
1.III.2. Existence and uniqueness of viscous approximations 77
Lemma 5 With the notation above and σ ∈ (−1,+∞), ρ0 = (σ)+ , the following holds.
(a) For all ρ ∈ IR+ and ε > 0 , σ 7→ Πε±(ρ; σ) do not decrease; nor do ρ0 7→ P±(ρ; ρ0) .
(b) For all ρ ∈ IR+ and ε > 0 , σ 7→ sign(ρ±−ρ0)Πε±(ρ; σ) do not increase; nor do
σ 7→ sign(ρ±−ρ0)P±(ρ; ρ0) .
(c) For all ε > 0 the maps σ 7→ Πε±(·; σ) are continuous for the L∞(IR+) topology; so
do σ 7→ P±(·; ρ0) .
(d) For all ε > 0 , the functions σ 7→ Sε±(σ) are continuous and strictly decreasing; so do
the functions ρ0 7→∫ ρ±
ρ0
√P±(r; ρ0)
dr
r.
Proof: The properties (a)-(c) for ε > 0 follow from Lemma 2. Hence (a)-(c) for P±
follow by Proposition 4 and Lemma 3(b). Now (b) and the Levi theorem yield (d). ⋄
Lemma 6 Let ρ±, u± be fixed.
(a) Assume (9)(a) holds. Then for all ε > 0 one has Sε±(σ) → −∞ as σ → +∞ .
(b) Assume (9)(b) holds. Then for all L > 0 there exists ε0(L) > 0 such that for all
0 < ε < ε0(L) one has limσ→+∞Sε±(σ) < −L .
(c) In both cases, one has
∫ ρ±
ρ0
√P±(r; ρ0)
dr
r→ −∞ as ρ0 → +∞ .
Proof: The proof is much the same as the one of Proposition 4 in Chapter 1.II.
(a) Assume, for instance, that S+(σ) is bounded from below by some constant −M ∈IR− . Set V0 := 1/(σ)+ , V+ := 1/ρ+ > V0 , and perform the transformation T−1p = f ,
T−1Πε+(·; σ) = Φ(·;V0) . We have Φ(·;V0) which is convex on [V0, V+] , Φ(V0;V0) =
f(V0) , Φ(V+;V0) = f(V+) , and Φ(·;V0) satisfies the equation (37) on (V0, V+) . Note that
f(V0) → −∞ as V0 → −∞ , by (9)(a). Therefore the convexity of Φ(·;V0) implies that
Φ(V+/2;V0) → −∞ as V0 → 0 ; on the other hand, it yields (V+/2)√d/dV Φ(V+/2;V0) ≤
−Sε+(σ) ≤ M independently of V0 ∈ (0, V+/2) . Hence we have a family of functions
Φ(·;V0)V0∈(0,V+/2) such that Φ(V+/2;V0) ↓ −∞ as V0 ↓ 0 , Φ(V+/2;V0) ≤ (2M/V+)2
uniformly in V0 , Φ(V+;V0) = f(V+) , and Φ(·;V0) satisfy (37) on (V+/2, V+) . Let us
compare Φ(·;V0) to the solution Ψ of the Cauchy problem Ψ(V+/2) = Φ(V+/2;V0) ,
Ψ(V+/2) = (2M/V+)2 + 1 for (37). By the maximum principle Lemma 2(b), which is valid
for the equation (37), we arrive to the conclusion that Φ(V+;V0) < Ψ(V+) , while this last
value can be made well-defined and strictly less than f(V+) for V0 small enough. This
contradiction proves (a).
For the proof of (c) in this case it is sufficient to pass to F±(·;V0) = T−1P±(·; ρ0) and
note that
∫ ρ±
ρ0
√P±(r; ρ0)
dr
r= −
∫ V±
V0
√d/dV F±(v;V0) dv . The last integrals diverge as
78 Riemann Problem for Gas Dynamics in Eulerian Coordinates
V0 ↓ 0 ; it is easy to see because F±(·;V0) are convex, F±(V0;V0) = f(V0) → −∞ and√d/dV F±(V ;V0) ≥ min1, d/dV F±(V ;V0) .
(b) First assume that f = T−1p has a finite number of points of inflexion. By Definition 1,
the function F+(·;V0) = T−1P+(·; ρ0) is the convex hull of f(·) on [V0, V+] = [1/ρ0, 1/ρ+] .
Its graph consists of a finite number of fragments of the graph of f(·) and of chords of this
graph. It follows easily by the Cauchy-Schwarz inequality that
∫ ρ+
ρ0
√P+(r; ρ0)
dr
r= −
∫ V+
V0
√d/dV F+(v;V0) dv ≤
≤ −∫ V+
V0
√d/dV f(v) dv =
∫ ρ+
ρ0
√p(r)
dρ
ρ→ −∞
as ρ0 → +∞ , by (9)(b). It follows by a density argument that
∫ ρ+
ρ0
√P+(r; ρ0)
dr
r→ −∞
as ρ0 → +∞ for a general flux function p(·) , which ends the proof of (c). Besides, the
claim of (b) follows now by Proposition 4 and Lemma 3(d). ⋄
Combining Propositions 1,2 and 3 and Lemmae 4,5,6, one easily deduces the main result
of this section.
Theorem 1 Let p(·) be continuous and strictly increasing on IR+ .
(a) Assume (9)(a) holds. Then for all ρ± > 0 , u± ∈ IR , ε > 0 there exists a unique
solution to the problem (1),(2) in the sense of Definition 2.
(b) Assume (9)(b) holds. Then for all ρ± > 0 , u± ∈ IR there exists ε0 = ε0(ρ±, u+−u−)such that for all 0 < ε < ε0 there exists a unique solution to the problem (1),(2) in
the sense of Definition 2.
Remark 2 At the present stage, it is not clear to the author whether the condition (9)(b)
suffices to have the existence of solutions to (1),(2) for all ε > 0 . On the other hand, it is
easy to see that if p(·) is locally absolutely continuous but fails to satisfy (9)(b), there is no
existence of a bounded self-similar solution to (1),(2) for arbitrary Riemann data. The exact
bound on u+ − u− in order to have the existence for ε in some non-empty neigbourhood of
0 is
−(u+ − u−) < limρ0→+∞
∫ ρ0
ρ+
√P+(r; ρ0)
dr
r+ lim
ρ0→+∞
∫ ρ0
ρ−
√P−(r; ρ0)
dr
r, (38)
where P±(·; ρ0) are the (-1)-hulls of p(·) on [ρ+, ρ0] and [ρ−, ρ0] , respectively. This
assertion follows from the extremal property of the upper (-1)-hull P (·) on [a, b] ⊂ (0,+∞) ,
which in fact maximizes the integral
∫ b
a
√Q(r)
dr
rwith respect to all Q(·) that satisfy the
assumptions (7)(i),(ii) in Definition 1.
1.III.3. Admissible solution of the problem (8),(2) 79
3 The admissible solution of the problem (8),(2)
In this section we study the global (with respect to data ρ±, u± ) solvability for the system
(8),(2) subject to the wave fan admissibility criterion (cf. [D74, D89, Tz96]).
Definition 3 Let (ρ, u) be a pair of functions such that ρ ∈ L∞(IR+ × IR; IR+) and
u ∈ L∞(ρ > 0; IR) , where ρ > 0 := (t, x) ∈ IR+ × IR | ρ(t, x) > 0 , ρ being an a.e.
defined representative of ρ . Then (ρ, u) is a wave-fan admissible solution of the problem
(8),(2), if
(i) the equations ρt + qx = 0 , qt + (e + p(ρ))x = 0 are fulfilled in D′(IR+ × IR) , where
q = ρu, e = ρu2 on the set ρ > 0 and q = 0, e = 0 on its complementary;
(ii) one has ess limt↓0
(‖ρ(t, ·)− ρ(0, ·)‖L1(−R,R) + ‖u(t, ·)− u(0, ·)‖L1(−R,R)
)= 0
for all R > 0 , where ρ(0, ·), u(0, ·) are given by (2);
(iii) in addition, there exists a sequence εnn∈IN ⊂ (0,+∞) , εn → 0 as n → ∞ , such
that (ρεn , uεn) tends to (ρ, u) as n → ∞ , where (ρεn , uεn) is a solution of (1),(2)
in the sense of Definition 2; more exactly, this means that ρεn → ρ , ρεnuεn → q and
ρεn(uεn)2 → e a.e. on IR+ × IR .
Note that, according to (i) of Definition 3, u remains undefined within the vaccuum state.
While (i),(ii) define a weak solution, (iii) presents an additional selection criterion which is
stronger than (i) unless p(·) degenerates to p(ρ) = const/ρ on some interval of IR+
depending on ρ±, u± . If such a degeneration does not occur, an infinity of weak solutions
that are not admissible can be observed (cf. [KrSt93]).
Due to (iii), a wave-fan admissible solution is actually self-similar. According to formulae
(40)-(45) and Theorem 2 below, it is unique and has the usual wave-fan structure. The
solution contains at most three “main” constant states: (ρ−, u−) at −∞ ; the vaccuum
state ρ ≡ 0 or a constant state in a neighbourhood of the unique point ξ0 such that there
exists limξ→ξ0 u(ξ) = ξ0 (in case there is no vaccuum state); and (ρ+, u+) at +∞ . These
states are separated by two wave fans of the first and the second family, respectively. Each
wave fan is a sequence of shocks, rarefactions, contact discontinuities and (in case p(·) is
not smooth) “rarefaction type” constant states. There is no vaccuum state in the solution
unless the intermediate “main” state is the one. In this case, u(ξ) − ξ → 0 as ξ enters
the vaccuum state from any side (cf. (41),(43) and (48) below). The necessary and sufficient
condition for vaccuum state to appear is that
u+ − u− ≥∫ ρ+
0
√P+(r; 0)
dr
r+
∫ ρ−
0
√P−(r; 0)
dr
r, (39)
80 Riemann Problem for Gas Dynamics in Eulerian Coordinates
where P±(·; 0) are the (-1)-hulls defined below. All these properties can be observed from
the following formulae for the wave-fan admissible solution, suggested by Propositions 2 and
4:
ρ(t, x) = ρ(x/t) =
[Ξ−]
−1(x/t), x/t < ξ+
[Ξ+]−1(x/t), ξ− < x/t
≡
[Ξ−]−1(x/t), x/t < ξ−
ρ0, ξ− < x/t < ξ+
[Ξ+]−1(x/t), ξ+ < x/t,
(40)
u(t, x) = u(x/t) =
U− [Ξ−]−1(x/t), x/t < ξ−
U−(ρ0) = U+(ρ0), ξ− < x/t < ξ+
(in case ρ0 > 0)
U+ [Ξ+]−1(x/t), ξ+ < x/t,
(41)
where
U±(ρ) := u± ∓∫ ρ±
ρ
√P±(r; ρ0)
dr
rfor ρ ∈ I(ρ0, ρ±), (42)
Ξ±(ρ) := U±(ρ)±√P±(ρ; ρ0) for ρ ∈ I(ρ0, ρ±), (43)
ξ± are defined by
ξ± := limρ∈I(ρ0,ρ±), ρ→ρ0
Ξ±(ρ) if ρ0 6= ρ±;
ξ− := −∞ and/or ξ+ := +∞ if ρ0 = ρ− and/or ρ0 = ρ+,
(44)
ρ0 = 0 in case (39) holds, and ρ0 ∈ (0,+∞) is the unique value that satisfies the relation
u+ − u− =
∫ ρ+
ρ0
√P+(r; ρ0)
dr
r+
∫ ρ−
ρ0
√P−(r; ρ0)
dr
r(45)
in case (39) fails; finally, P±(·; ρ0) are the (-1)-hulls of the graph of p(·) on I(ρ0, ρ±) ,
respectively, as defined in Definition 1.
Let state the main result of this chapter.
Theorem 2 Assume that continuous strictly increasing function p(·) satisfies (9). Let ρ± >
0 , u± ∈ IR . Then the solution (ρε, uε) to the problem (8),(2) (which exists and is unique, at
least for ε sufficiently small) tends as ε ↓ 0 , in the sense of Definition 3(iii), to (ρ, u) given
by the formulae (40)-(45). The pair (ρ, u) is the unique solution of the Riemann problem
(8),(2) in the sense of Definition 3.
Proof: According to Theorem 1, there exists a unique solution to (1),(2), at least for
ε sufficiently small; let denote this solution by (ρε, uε) . By Propositions 1 and 2 (ρε, uε)
corresponds to some σε ∈ (−1,+∞) and Πε±(·; σε) such that (22)-(25) and (26)-(30)
hold. Consider the set (σε)+ ⊂ IR+ ; it has an accumulation point ρ0 in IR+ . Choose
1.III.3. Admissible solution of the problem (8),(2) 81
a sequence εn ↓ 0 such that σεn → σ0 as n → ∞ ; let omit the subscript n , because
we will prove later that σ0 does not depend on any particular sequence of ε ↓ 0 . Consider
separately the three possibilities: ρ0 ∈ (0,+∞) , ρ0 = +∞ and ρ0 = 0 .
a) ρ0 ∈ (0,+∞) . As in Chapter 1.II, from Proposition 4 and Lemma 5(a),(c) it follows
that
Πε±(·; σε) → P±(·; ρ0) in L∞(IR+) as ε → 0, (46)
where Πε±, P± are extended to IR+ as in Lemma 5. Similarly, extend Uε
± , Uε in (28)
and (42) by constants to continuous functions on IR+ . By Lemma 3(d), Uε±(·) converge to
U±(·) uniformly on I(ρ0, ρ±) , respectively. Since (25) and (45) write also as Uε−((σ
ε)+) =
Uε+((σ
ε)+) and U−(ρ0) = U+(ρ0) , one deduces (45). It follows by Lemma 5 that ρ0 is
uniquely determined by p(·) , ρ± and u± in case a).
Further, by Lemma 3(c) and (43),(44) one has Ξ±(ρ1) ≤ ξ− ≤ U−(ρ0) = U+(ρ0) ≤ξ+ ≤ Ξ+(ρ2) for all ρ1 ∈ I(ρ0, ρ−) , ρ2 ∈ I(ρ0, ρ+) . Considering separately the cases
ρ0 > ρ+, ρ0 = ρ+ , ρ0 < ρ+ , one deduces by Lemma 3(e) from (46),(26) and (40) that
ρε(·) → ρ(·) a.e. on (ξ−,+∞) . Similarly, one gets ρε(·) → ρ(·) a.e. on (−∞, ξ+) , so
that the convergence actually takes place for a.a. ξ ∈ IR . Consequently, uε(·) → uε(·) a.e.
on IR .
b) ρ0 = +∞ . Actually, this case is impossible. Indeed, for all L > 0 we have
u+ − u− =
∫ ρ+
σε
√Πε
+(r; σε)dr
r+
∫ ρ−
σε
√Πε
−(r; σε)dr
r≥
≥∫ ρ+
L
√Πε
+(r;L)dr
r+
∫ ρ−
L
√Πε
−(r;L)dr
r
for ε small enough, by Lemma 5(d). Passing to the limit as ε→ 0 , we obtain
u+ − u− ≥∫ ρ+
L
√P+(r;L)
dr
r+
∫ ρ−
L
√P−(r;L)
dr
r
by Proposition 4 and Lemma 3(d). By Lemma 6(c), this last quantity tends to −∞ as
L→ +∞ , provided (9) holds. Thus u+ − u− = −∞ , which is contradictory.
c) ρ0 = 0 . As in case a), one has (46). Therefore Πε±(·; σε) → P±(·; 0) a.e. on (0, ρ±) ,
respectively, by Lemma 3(b). It follows by the Fatou Lemma that
u+ − u− = lim infε→0
(∫ ρ+
0
√Πε
+(r; σε)dr
r+
∫ ρ−
0
√Πε
−(r; σε)dr
r
)≥
≥∫ ρ+
0
√Πε
+(r; 0)dr
r+
∫ ρ−
0
√Πε
−(r; 0)dr
r.
(47)
Note that, since c) is excluded and since we have seen that in case a) this last quantity is
necessarily greater than u+ − u− , the accumulation point ρ0 is always unique and finite.
Let prove that ρε → ρ , ρεuε → ρu and ρε(uε)2 → ρu2 on the set ξ | ρ(ξ) > 0 , and
that ρε → 0 , ρεuε → 0 and ρε(uε)2 → 0 on its complementary.
82 Riemann Problem for Gas Dynamics in Eulerian Coordinates
First assume (47) holds with equality. Then we still have U−(0) = U+(0) . Moreover,
P (ρ; 0) → 0 as ρ→ 0 (48)
whenever vaccuum appears. Indeed, one can show as in the proof of Lemma 3(c) that
Ξ±(ρ) = u± ∓∫ ρ±
ρ
√P±(r; 0) ±
√P±(ρ; 0)
dr
rare monotone on (0, ρ±) ; on the other
hand,
∫ ρ±
ρ
√P±(r; 0)
dr
rare also monotone and converge as ρ ↓ 0 . Therefore there exist
limits as ρ ↓ 0 of
√P±(ρ; 0) , which are necessarily zero. Thus Ξ−(ρ1) ≤ ξ− = U−(0) =
U+(0) = ξ+ ≤ Ξ+(ρ2) for all ρ1 ∈ [0, ρ−] , ρ2 ∈ [0, ρ+] ; moreover, ρ(ξ) > 0 for all
ξ 6= ξ− = ξ+ . The convergence of ρε, uε to ρ, u , respectively, a.e. on IR follows as in case
a).
Secondly, assume that the inequality in (47) is strict. By (48) we have Ξ−(ρ1) ≤ ξ− =
U−(0) < U+(0) = ξ+ ≤ Ξ+(ρ2) , with ρ1 , ρ2 as above. One more time, we deduce
the convergence of ρε, uε to ρ, u , respectively, a.e. on (−∞, ξ−) ∪ (ξ+,+∞) . Note
that Ξ±(ρ) are strictly monotone at ρ = 0 , since P±(·; 0) are strictly increasing and
P±(+0; 0) = 0 . It follows that ρ(ξ) → 0 as ξ ↑ ξ− or ξ ↓ ξ+ . For ε sufficiently small, ρε
has no points of maximum on IR , so that for all δ > 0 we infer maxξ∈[ξ−−δ,ξ++δ] ρε(ξ) =
maxρε(ξ− − δ), ρε(ξ+ + δ) . By the last observation, it follows that ρε → 0 uniformly
on [ξ−, ξ+] . Besides, uε is nondecreasing on IR for ε sufficiently small, hence uniformly
bounded, so that ρεuε → 0 and ρε(uε)2 → 0 on [ξ−, ξ+] .
Thus we have shown that (ρ, u)(·, ·) satisfies Definition 3(i),(iii). Besides, (ρ, u)(±∞) =
(ρ±, u±) by (40)-(42). Since, in addition, ρ(·) , u(·) are both monotone at ±∞ , Defini-
tion 3(ii) holds as well. ⋄
Remark 3 In the equations of isentropic gas dynamics u represents the velocity in gas and
has no physical meaning inside the vaccuum state, while the specific impulse q and specific
energy e are, naturally, both zero within vaccuum. Nevertheless, one could ask for a limit
of uε even inside the vaccuum state. In case p ∈ W 1,1(0, R) for some R > 0 and
limρ↓0 p(ρ) = 0 , we are able to prove that the unique limit of uε inside the vaccuum state
is the identity function u(ξ) = ξ . This is due to the formula (43) and the uniform in ε
convergence of√
Πε±(ρ; 0) to 0 as ρ ↓ 0 . In turn, this last property results from (48),
Proposition 4, Lemma 3(b) and the following kind of maximum principle:
maxρ∈[0,δ]
Πε±(ρ; 0) = maxΠε
±(δ; 0), supρ∈(0,δ)
p(ρ) for δ small enough,
which can be easily deduced from the equation (22). In general, we ignore whether the
convergence always take place to the same limit u(ξ) ≡ ξ within the vaccuum state; but we
still observe that u(ξ) coincides with ξ on its boundary.
CHAPTER 1.IV
L1 -Theory of Scalar Conservation Law
with Continuous Flux Function†
Introduction
We consider the Cauchy problem
∂u
∂t+ divx φ(u) = g on Q = (t, x); t ∈ (0, T ), x ∈ IRN
u(0, ·) = f on IRN(CP )
where φ : IR 7→ IRN is only assumed to be continuous and (f, g) satisfy
f = f0 + c with c ∈ IR, f0 ∈ L1(IRN) ∩ L∞(IRN),
g ∈ L1(Q), g(t, ·) ∈ L∞(IRN) for a.a. t ∈ (0, T ) and
∫ T
0
‖g(t, ·)‖∞dt <∞.(1)
A solution of (CP) will be understood in the sense of the generalized entropy solution
(g.e.s.) as introduced by S.N.Kruzhkov (cf. [K69a, K69b, K70a]). In the case of a locally
Lipschitz continuous flux function φ , there exists a unique bounded g.e.s.; this is actually
true for any (f, g) satisfying
f ∈ L∞(IRN), g ∈ L1loc(Q),
g(t, ·) ∈ L∞(IRN) for a.a. t ∈ (0, T ) and
∫ T
0
‖g(t, ·)‖∞dt <∞.(2)
For the general continuous flux function φ the situation is more delicate. Let consider the
particular case N = 2 , φ(u) =(
|u|α−1uα
, |u|β−1uβ
). It has been shown in [KP90] that if
α 6= β , α + β < 1 , then for some f ∈ L∞(IR2) the problem (CP), with g = 0 , has
a one-parameter family of different bounded g.e.s.. On the other hand, it has been shown
†This chapter will be published in [BABK].
84 Scalar Conservation Law with Continuous Flux Function
in [BK96] that if (f, g) satisfy (1), then for any α, β > 0 there exists a unique bounded
g.e.s. of (CP). In this chapter we shall improve this last result, showing (cf. Theorem 3) that
(CP) has a unique bounded g.e.s. for any (f, g) satisfying (1) according to whether the flux
function φ satisfies
There exist orthonormal vectors ξ1, . . . , ξN−1 such that
r ∈ IR 7→ ξi · φ(r) ∈ IR is nondecreasing , i = 1, . . . , N − 1.(3)
Actually, while we shall prove another uniqueness result (cf. Theorem 4), we still do not
know whether there is or is not uniqueness of bounded g.e.s. for (f, g) satisfying (1) with
any continuous flux function φ ; but we shall prove that there always exist a maximum and
a minimum bounded g.e.s. of (CP). More precisely, for any continuous flux function φ and
(f, g) satisfyingf = f0 + c with c ∈ IR, f0 ∈ L∞
0 (IRN) =h ∈ L∞(IRN); λN|h| > δ <∞† ∀δ > 0
g ∈ L1loc(Q), g(t, ·) ∈ L∞
0 (IRN) for a.a. t ∈ (0, T ) and
∫ T
0
‖g(t, ·)‖∞dt <∞,(4)
for all c ∈ IR there exist a maximum and minimum bounded g.e.s. of (CP), which coincide
except for a countable set of values of c depending on φ , f0 and g (cf. Theorem 1 and
Proposition 1).
As pointed out in [C72] and [B72], solutions of (CP) for (f, g) satisfying (1) can be
constructed through the nonlinear semigroup theory from the solutions of the equation
u+ divx φ(u) = f on IRN . (E)
As was done in [BK96], we shall derive for the equation (E) the same properties as for the
Cauchy problem (CP); actually we shall prove (cf. Corollary 1), for φ and c given, that
there is uniqueness of a bounded g.e.s. of (CP) for all (f, g) satisfying (1) if and only if there
is uniqueness of a bounded g.e.s. of (E) for all f = f0 + c with f0 ∈ L1(IRN) ∩ L∞(IRN) .
1 Existence of maximum and minimum generalized
entropy solutions
Throughout this chapter φ : IR 7→ IRN is a continuous function and we consider the Cauchy
problem (CP) as well as the equation (E). Recall the following definition:
Definition 1 Let f ∈ L1loc(IR
N) . A sub-g.e.s. (generalized entropy subsolution) (respec-
tively, super-g.e.s.) of (E) is a function u ∈ L∞loc(IR
N) satisfying
α · (u− k) + divx
α · (φ(u)− φ(k))
≤ α · (f − k) in D′(IRN) for any k ∈ IR,
† λN denote the N-dimensional Lebesgue measure; |h| > δ stands for x ∈ IRN ; |h(x)| > δ and
so on.
1.IV.1. Existence of maximum and minimum g.e.s. 85
where α = sign+(u − k) ‡ (respectively, sign−(u − k) ). A function u is a generalized
entropy solution (g.e.s.) of (E) if it is both sub- and super-g.e.s.
Definition 2 Let f ∈ L1loc(IR
N) and g ∈ L1loc(Q) . A sub-g.e.s. (respectively, super-g.e.s.)
of (CP) is a function u ∈ L∞loc(Q) satisfying
∂
∂t
α · (u− k)
+ divx
α · (φ(u)− φ(k))
≤ α · g in D′(Q) for any k ∈ IR,
where α = sign+(u−k) (respectively, sign−(u−k) ), and (u(t, ·)−f)+ → 0 (respectively,
(u(t, ·)− f)− → 0 ) in L1loc(IR
N ) as t → 0 essentially. A function u is a g.e.s. of (CP) if
it is both sub- and super-g.e.s.
The main result is the following theorem.
Theorem 1 Let (f, g) satisfy (4). Then there exist a maximum and a minimum bounded
g.e.s. of (E) and of (CP).
More precisely, considering the equation (E) and f = f0+c with c ∈ IR , f0 ∈ L∞0 (IRN) ,
we shall prove that there exists a (unique) g.e.s. u ∈ L∞(IRN) such that u ≥ u a.e. on
IRN for any sub-g.e.s. u ∈ L∞(IRN ) of (E).
This g.e.s. u will be obtained as the a.e. pointwise limit of a nonincreasing sequence
un , where un is any bounded g.e.s. of (E) corresponding to f = f0+ cn with a sequence
cn in IR decreasing to c .
The same corresponding results are valid for (CP) and minimum solutions.
The main new ingredient in the proof of Theorem 1 is the following lemma.
Lemma 1 a) Let u and u be bounded sub- and super-g.e.s., respectively, of (E), corre-
sponding to f and f ∈ L1loc(IR
N) , respectively. Assume that
λNx ∈ IRN ; u(x) > u(x)
<∞, (5)
then∫(u− u)+ +
∫
u>u
(f − f)− ≤∫
u≥u
(f − f)+, (6)
and in particular, if f ≤ f a.e. on u ≥ u , then u ≤ u a.e. on IRN .
b) Let u and u be bounded sub- and super-g.e.s., respectively, of (CP), corresponding
to (f, g) and (f , g) ∈ L1loc(IR
N)× L1loc(Q) , respectively. Assume that
λN+1
(t, x) ∈ Q; u(t, x) > u(t, x)
<∞, (7)
‡We use the notation sign+ for the Heaviside function, i.e. the characteristic function of (0,+∞) , and
sign−(r) = −sign+(−r) .
86 Scalar Conservation Law with Continuous Flux Function
then for a.a. t ∈ (0, T )∫
(u(t, ·)− u(t, ·))+ +
∫ t
0
∫
u>u
(g − g)− ≤∫(f − f)+ +
∫ t
0
∫
u≥u
(g − g)+, (8)
and in particular, if f ≤ f a.e. on IRN and g ≤ g a.e. on u ≥ u , then u ≤ u a.e. on
Q .
Proof: Applying Lemma 3.1a) in [BK96] (cf. also [K70a, B72, C72, Ba88]), we have in the
case a)∫
(u− u)+ζ +
∫
u>u
(f − f)−ζ ≤∫
u≥u
(f − f)+ζ +
∫|φ(u)− φ(u)|χu>u |Dζ |
for all ζ ≥ 0 , ζ ∈ D(IRN) . By the assumption (5), |φ(u) − φ(u)|χu>u ∈ L1(IRN) , so
that we may let ζ tend to 1 to obtain (6) at the limit and prove a). The proof of b) is
identical using Lemma 3.1b) in [BK96] and (7). ⋄We also need the following general existence result, partially contained in [B72] and [KH74],
for which we give a complete proof in the Appendix.
Lemma 2 Let f ∈ L∞(IRN) (resp., and g ∈ L1loc(Q) satisfying g(t, ·) ∈ L∞(IRN ) for a.a.
t ∈ (0, T ) and
∫ T
0
‖g(t, ·)‖∞dt < ∞ ). Then there exists a bounded g.e.s. of (E) (resp.
(CP)).
Proof of Theorem 1: Let cn be a sequence in IR decreasing to c and, for n ∈ IN ,
un be a bounded g.e.s of (E) corresponding to fn = f0+ cn . Such a g.e.s. exists by Lemma
2.
Fix n > m . Set h = cn+cm2
and take 0 < δ < cm−cn2
; δ = cm−cn2
−α for some α > 0 .
Using Theorem 2.2a) in [BK96], we have∫(un−um+2δ)+ ≤
∫(un−h+δ)+ +
∫(h+δ−um)+ =
∫ ((un−cn)−α
)++
+
∫ ((um−cm)+α
)−≤∫(f0−α)+ +
∫(f0+α)
− =
∫(|f0| − α)+ <∞;
it follows that |un > um| <∞ and thus we deduce from Lemma 1a) that un ≤ um .
Define u = limn→∞
un ; this is, clearly, a bounded g.e.s. of (E) corresponding to f = f0+c .
Let now u be a bounded sub-g.e.s. of (E); with the same argument as above, u ≤ un a.e.
for all n and thus u ≤ u a.e. In other words, u is the maximum bounded g.e.s. of (E).
The proof of existence of the maximum bounded g.e.s. of (CP) is similar using Lemma
1b); considering a bounded g.e.s. un of (CP) corresponding to (fn, g) , we only need to
show that
supt∈[0,T ]
[∫ ((un(t)−cn)−α
)++
∫ ((um(t)−cm)+α
)−]<∞. (9)
1.IV.2. L1 semigroup approach 87
To prove (9), for M > 0 set κM (t) = inf
κ ;
∫(|g(t)| − κ)+ ≤M
. We have
κM(t) ≤ ‖g(t)‖∞ and thus κM ∈ L1(0, T ) ; on the other hand, for a.a. t ∈ (0, T ) ,
since g(t) ∈ L∞0 (IRN) , κM(t) decrease to 0 as M increase to ∞ . Thus there exists
M > 0 such that
∫ T
0
κM(t)dt ≤ α/2 . Fix M such that this is satisfied, and set kM(t) =
α/2 +
∫ t
0
κM(s)ds for t ∈ [0, T ] . Applying Theorem 2.2b) from [BK96], we get for all
t ∈ [0, T ]∫ (
(un(t)−cn)−α)+
+
∫ ((um(t)−cm)+α
)−≤∫ (
(un(t)−cn)−kM(t))+
+
+
∫ ((um(t)−cm)+kM(t)
)−≤∫ (
|f0|−α/2)+
+
∫ t
0
∫ (|g(s)|−κM(s)
)+ds ≤
≤∫ (
|f0|−α/2)+
+MT.
The proof of existence of the minimal bounded g.e.s. for (E) and (CP) is identical. ⋄We actually do not know whether there is in general uniqueness of a bounded g.e.s. of (E)
or (CP). However, we can prove the following result.
Proposition 1 Let f0 ∈ L∞0 (IRN) (resp., and g ∈ L1
loc(Q) , g(t, ·) ∈ L∞0 (IRN ) for a.a.
t ∈ (0, T ) and
∫ T
0
‖g(t, ·)‖∞dt < ∞ ). Then there exists an at most countable set N in
IR such that for all c ∈ IR \ N the equation (E) (resp., the problem (CP)) with f = f0 + c
has a unique bounded g.e.s..
Proof: For c ∈ IR , denote by u(c) (resp., u(c) ) the maximum (resp. minimum) bounded
g.e.s. By the proof above, we know that c 7→ u(c) and c 7→ u(c) are nondecreasing from
IR into L∞ continuous from the right and the left, respectively, for the L1loc topology in
L∞ ; moreover, for c1 < c2 , u(c1) ≤ u(c1) ≤ u(c2) . Thus it follows that u(c) = u(c) a.e.
for any c except an at most countable set in IR . ⋄
2 The L1 semigroup approach
In this section, using the nonlinear semigroup theory in L1 , we make the relation between
the equation (E) and the problem (CP) under the assumption (1) on the data (f, g) clearer.
For simplicity we shall assume c = 0 .
For λ > 0 and f ∈ L1(IRN) ∩ L∞(IRN) , the equation
u+ divx λφ(u) = f in IRN (10)
has a maximum bounded g.e.s. that we shall denote by J+λ f ; by Corollary 2.1 in [BK96]
J+λ f ∈ L1(IRN ) . In other words, J+
λ maps L1(IRN ) ∩ L∞(IRN) into itself. Let us start
with the following results.
88 Scalar Conservation Law with Continuous Flux Function
Proposition 2 With the notation above, the following properties hold:
1. for any λ > 0 , J+λ is a T-contraction for the L1 -norm, i.e.
∫(J+
λ f − J+λ f)
+ ≤∫
(f − f)+ ∀f, f ∈ L1(IRN) ∩ L∞(IRN );
2. J+λ λ>0 is a resolvent family, i.e.
J+λ f = J+
µ
(µ
λf +
λ− µ
λJ+λ f
)∀λ, µ > 0, f ∈ L1(IRN) ∩ L∞(IRN);
3. the range R(J+λ ) , independent of λ by 2) , is dense in L1(IRN) .
Proof: For Part 1), let f, f ∈ L1(IRN ) ∩ L∞(IRN) and, for δ > 0 , denote by uδ, uδ
bounded g.e.s. of (E) corresponding to f + δ, f + δ respectively. As in the proof of Theorem
1, we have λNuδ > u2δ <∞ and then, by Lemma 1,
∫(uδ − u2δ)+ ≤
∫(f − f − δ)+ ≤
∫(f − f)+ . At the limit as δ → 0 , uδ → J+
λ f and u2δ → J+λ f a.e., so that we get
Part 1) by the Fatou Lemma.
For Part 2), let f ∈ L1(IRN) ∩ L∞(IRN) and assume first that λ > µ > 0 . Set
u = J+λ f ; it is a bounded g.e.s. of
v + divx µφ(v) =µ
λf +
λ− µ
λu
and so u ≤ v = J+µ
(µλf + λ−µ
λu). Then µ
λf + λ−µ
λu ≤ µ
λf + λ−µ
λv and v is a bounded
sub-g.e.s. of u + divx λφ(u) = f . We deduce v ≤ u and thus v = u . To complete the
proof of Part 2), we apply the abstract Lemma 3 below.
For the proof of Part 3), let f ∈ L1(IRN) ∩ L∞(IRN) and, for λ > 0 , set uλ = J+λ f .
We have uλ ∈ R(J+λ ) and this set is clearly, by Part 2) , independent of λ . Since ‖uλ‖∞ ≤
‖f‖∞ (see Corollary 2.1 in [BK96]), it follows immediately that uλ → f in D′(IRN) as
λ→ 0 ; indeed, being a g.e.s., uλ is also a solution of (10) in the sense of distributions. Now
using translation invariance and Part 1),
∫|uλ(x+h)−uλ(x)|dx ≤
∫|f(x+h)− f(x)|dx ,
so that the set uλλ>0 is relatively compact in L1loc(IR
N) and uλ → f in L1loc(IR
N) . At
last ‖uλ‖1 ≤ ‖f‖1 (cf. Corollary 2.1 in [BK96]) so that uλ → f in L1(IRN) ; indeed, for
any compact set K in IRN we have
lim supλ→0
‖uλ − f‖1 ≤ lim supλ→0
(∫
K
|uλ − f |+∫
|uλ| −∫
K
|uλ|+∫
IRN \K
|f |)
≤
≤∫
|f | −∫
K
|f |+∫
IRN\K
|f | = 2
∫
IRN\K
|f |,
which can be made as small as we want. ⋄
1.IV.2. L1 semigroup approach 89
Lemma 3 Let X0 be a linear subspace of a Banach space X and Jλλ>0 be a family of
non-expansive mappings from X0 into X0 . If the resolvent identity Jλ = Jµ(µλI + λ−µ
λJλ)
holds for all 0 < µ < λ , then it still holds for any λ, µ > 0 .
Proof: Following [BCP], Exercise E8.2, denote by Aλ the multivalued operator from X0
into itself defined by
v ∈ Aλu ⇔ u, v ∈ X0, u = Jλ(u+ λv);
the graph of this operator is(Jλf,
f−Jλfλ
); f ∈ X0
and one has (I + λAλ)
−1 = Jλ .
For given λ, µ > 0 , the equality Jλ = Jµ(µλI + λ−µ
λJλ)
is equivalent to the inclusion
Jλ ⊂ Jµ(µλI + λ−µ
λJλ)
since these two maps are everywhere defined in the linear space X0 ;
so it is also equivalent to the inclusion Aλ ⊂ Aµ .
By assumption Aλ ⊂ Aµ for 0 < µ < λ . We deduce that for any λ > 0 , Aλ is
an accretive operator; indeed, for µ > 0 small enough ( µ < λ ), (I + µAλ)−1 is a non-
expansive mapping since it is contained in Jµ . Thus, for 0 < λ < µ , (I + µAλ)−1 is a
single-valued operator in X0 containing (I + µAµ)−1 = Jµ , which is everywhere defined in
X0 ; so (I + µAλ)−1 = (I + µAµ)
−1 and Aλ = Aµ . ⋄
As we have seen in the proof above, there exists a multivalued operator A+ in L1(IRN )∩L∞(IRN ) such that J+
λ = (I + λA+)−1 for any λ > 0 . This operator is accretive densely
defined in L1(IRN) and R(I + λA+) = D(J+λ ) = L1(IRN)∩L∞(IRN) is dense in L1(IRN)
for any λ > 0 . This operator A+ is exactly defined by
v ∈ A+u ⇔ ∃f ∈ L1(IRN) ∩ L∞(IRN) such that
u is the maximum bounded g.e.s. of (E) and v = f − u;
it follows that A+ is actually single-valued since for v ∈ A+u one has v = divxφ(u) in
D′(IRN) . By the Crandall-Liggett theorem (cf. [CL71, B72, C76, BCP]) for any (f, g) ∈L1(IRN)×L1(Q) there exists a unique mild (or integral) solution u ∈ C([0, T ];L1(IRN )) of
du
dt+ A+u = g on (0, T ), u(0) = f. (11)
Theorem 2 With the notations above, for (f, g) satisfying (1) with c = 0 , the mild solution
of (11) is the maximum bounded g.e.s. of (CP).
Proof: With the same argument as in [C72] and [B72], it is clear that, under the assumptions
(1), the mild solution u ∈ C([0, T ];L1(IRN )) of (11) is in L∞(Q) and a g.e.s. of (CP).
Therefore u ≤ u a.e. on Q .
Now we prove that u satisfies∫(u(t)− w)+ ≤
∫(f − w)+ +
∫ t
0
[u(τ)− w, g(τ)− A+w
]+dτ (12)
90 Scalar Conservation Law with Continuous Flux Function
for a.a. t ∈ (0, T ) and for all w ∈ D(A+) , where for u, f ∈ L1(IRN)
[u, f
]+=
∫
u>0
f +
∫
u=0
f+ ≡ infµ>0
∫(u+ µf)+ −
∫u+
µ.
Using translation invariance in time, we shall have ddt
∫(u(t)−w)+ ≤
[u(t)−w, g(t)−A+w
]+
in D′((0, T )) . Applying the results of [BaB92], we shall conclude that u ≤ u a.e. on Q
and this will end the proof.
Let w ∈ D(A+) , δ > 0 . By definition w = J+1 h , A+w = h− J+
1 h with some
h ∈ L1(IRN)∩L∞(IRN) . Consider wδ a bounded g.e.s. of w+divx φ(w) = h+δ . Take wδ
as a stationary bounded g.e.s. of the corresponding (CP); since wδ − δ ∈ L∞(0, T ;L1(IRN))
and u ∈ L∞(0, T ;L1(IRN)) (cf. Corollary 2.1 in [BK96]), we have λN+1(u > wδ
)<∞
with the same argument as in the proof of Theorem 1. Thus Lemma 1b) yields
∫ (u(t)− wδ
)+≤
≤∫ (
f − wδ)+
+
∫ t
0
[u(τ)− wδ, g(τ)− (h+δ−wδ)
]+dτ ≤
∫ (f − wδ
)++
+
∫ t
0
1
µ
∫ (u(τ)− wδ + µ(g(τ)−A+w+wδ−w−δ)
)+−∫ (
u(τ)− wδ)+
dτ
(13)
for any µ > 0 . As δ decreases to 0 , wδ decreases to w ; moreover for 0 < µ ≤ 1 ,(u(τ)− wδ + µ(g(τ)−A+w+wδ−w−δ)
)+increases to
(u(τ)− w + µ(g(τ)− A+w)
)+.
So we may pass to the limit in (13) and obtain
∫ (u(t)− w
)+≤
≤∫ (
f − w)+
+
∫ t
0
1
µ
∫ (u(τ)− w + µ(g(τ)− A+w)
)+−∫ (
u(τ)− w)+
dτ
for any 0 < µ ≤ 1 . Letting µ → 0 yields (12). ⋄
Remark 1 Of course, one may consider minimum bounded g.e.s. of (10), define the corre-
sponding operator A− , and prove the following result analogous to Theorem 2:
The mild solution of (11) with A− in place of A+ is exactly the minimum bounded
g.e.s. of (CP).
Corollary 1 For a given continuous flux function φ and c ∈ IR , the following assertions are
equivalent:
(i) for all f = f0 + c with f0 ∈ L1(IRN) ∩ L∞(IRN ) there exists a unique bounded
g.e.s. of (E);
(ii) for all (f, g) satisfying (1) there exists a unique bounded g.e.s. of (CP).
1.IV.3. Uniqueness results in L1(IRN) ∩ L∞(IRN) 91
Proof: Replacing φ(r) by φ(r + c) , we may assume c = 0 .
If (i) holds, the operators A+ and A− coincide and then by Theorem 2, (see also
Remark 1), for any (f, g) satisfying (1) the maximum and minimum bounded g.e.s. of (CP)
coincide, so that (ii) holds.
Conversely, assume that (ii) holds and for f ∈ L1(IRN) ∩ L∞(IRN) let u, u be two
bounded g.e.s. of (E). Then u(t) ≡ u is a bounded g.e.s. of (CP) corresponding to (u, g(t)≡f−u) and so, by uniqueness, it is the maximum bounded g.e.s. and then, by Theorem 2,
it is the unique mild solution of the corresponding evolution problem (11). In the same way
u(t) ≡ u is the unique mild solution of (11) corresponding to (u, g(t)≡f−u) . Then by the
integral inequality (see [B72, BCP, BW94])
−∫
|u− u| =[u(t)− u(t), (f − u)− (f − u)
]≥ d
dt
∫|u(t)− u(t)| = 0,
where [·, ·] stands for the bracket associated with the standard norm in L1 , i.e., for all
u, f ∈ L1(IRN ) ,[u, f
]=
∫
u 6=0
f sign u+
∫
u=0
|f | . It follows that u = u a.e. in IRN
so that (i) holds. ⋄
Remark 2 For f = f0 + c with f0 ∈ L1(IRN) ∩ L∞(IRN) , any bounded g.e.s. of (E) is
in c + L1(IRN) (cf. Corollary 2.1 in [BK96]); so there is uniqueness of a bounded g.e.s. to
(E) if and only if
∫ (u(f)− u(f)
)= 0 , where u(f) and u(f) are the maximum and the
minimum bounded g.e.s. of (E), respectively.
By Part 1) of Proposition 2, for given c ∈ IR the map f0 7→ u(f0+c)−c is a contraction
for the L1 -norm; the same holds for f0 7→ u(f0+c)−c so that f0 7→∫ (
u(f0+c)−u(f0+c))
is continuous for the L1 -topology. It follows that (i) of Corollary 1 is equivalent to the
uniqueness of a bounded g.e.s. of (E) for all f0 in some L1 -dense subset of L1(IRN ) ∩L∞(IRN ) .
Consequently, since the L1 -topology in L1(IRN) ∩ L∞(IRN ) is separable, Proposition 1
can be improved as follows.
Proposition 3 There exists an at most countable set N in IR such that, for all c ∈ IR\N ,
the two properties (i) and (ii) of Corollary 1 hold.
3 Some uniqueness results in L1(IRN) ∩ L∞(IRN)
As noted in the introduction, we still do not know if, for any continuous flux function φ , there
is uniqueness of a bounded g.e.s. to (CP) under assumption (1) or to (E) for all f = f0 + c ,
f0 ∈ L1(IRN) ∩ L∞(IRN) , c ∈ IR . In this section we shall improve some uniqueness results
shown in [BK96].
92 Scalar Conservation Law with Continuous Flux Function
Theorem 3 Assume there exist orthonormal vectors ξ1, . . . , ξN−1 and C : IR → [0,+∞)
continuous such that
d
drξi · φ(r) ≤ C(r) in D′(IR) for i = 1, . . . , N − 1. (14)
Then for any c ∈ IR the two properties of Corollary 1 hold.
We shall need the following lemma.
Lemma 4 Let ξ ∈ IRN , ξ 6= 0 , such that r ∈ IR 7→ ξ · φ(r) is nondecreasing. Let α ∈ IR
and f ∈ L∞(IRN) with support contained in H = x ∈ IRN ; ξ · x ≥ α. Assume that one
of the following conditions holds:
a) there exists a unique bounded g.e.s. of (E);
b) f ∈ L∞0 (IRN) .
Then for all bounded g.e.s. u of (E) the support of u is also contained in H .
Proof: a) This is clearly true if φ is locally Lipschitz continuous. Indeed, by the definition
of g.e.s.,
∫|u(x)|ρ
(α− ξ · x
ε
)ζ(x)dx ≤
∫|f(x)|ρ
(α− ξ · x
ε
)ζ(x)dx+
+
∫sign u(x)(φ(u(x))− φ(0))
−ξερ′(α− ξ · x
ε
)ζ(x) + ρ
(α− ξ · x
ε
)Dζ(x)
dx
for ζ ∈ D(IRN) , ζ ≥ 0 , ρ ∈ C∞(IR) with ρ′ ≥ 0 , ρ = 0 on (−∞, 0] , ρ = 1 on
[1,+∞) , and ε ≥ 0 . Since sign u(x)(φ(u(x)) − φ(0)) · ξ ≥ 0 and f(x)ρ(α−ξ·x
ε
)≡ 0 ,
using the Lipschitz continuity of φ we get
∫|u(x)|ρ
(α− ξ · x
ε
)ζ(x)dx ≤ C
∫|u(x)|ρ
(α− ξ · x
ε
)|Dζ(x)|dx.
Let ε → 0 and ζ → 1 ; it follows that u = 0 a.e. in IRN \H (see, for instance, Lemma
1.1 in [BK96]).
For the general case, let φn = φ ∗ ρn , where ρn is a sequence of mollifiers, and let un
be the bounded g.e.s. of (E) corresponding to the flux φn . Using the contraction property
and translation invariance, we see that the sequence un is relatively compact in L1loc(IR
N) ;
clearly any limit point is a bounded g.e.s. of (E) and then by the uniqueness assumption u is
the limit in L1loc(IR
N) of the sequence un . Note that r ∈ IR 7→ ξ ·φn(r) is nondecreasing
for all n ; thus by the argument above supp un ⊂ H and the same is true at the limit.
b) Let f ∈ L∞0 (IRN) . By Theorem 1 the equation (E) has a maximum bounded g.e.s.
u , which is the limit of any sequence un of bounded g.e.s. of (E) corresponding to
fn = f + cn with cn ↓ 0 . Moreover, by Proposition 1 we may choose cn so that there
is uniqueness of bounded g.e.s. of (E) corresponding to fn . By the first part of Lemma 4,
1.IV.3. Uniqueness results in L1(IRN) ∩ L∞(IRN) 93
supp (un − cn) ⊂ H , therefore supp u ⊂ H . Using the same argument for the minimum
bounded g.e.s. u of (E), we see that the conclusion of Lemma 4 still holds. ⋄
Proof of Theorem 3: Replacing φ(r) by φ(r+ c)− φ(c) , we may assume c = 0 and
φ(0) = 0 .
Since we are working with bounded solutions, we may also assume that C(r) is constant.
By replacing ξi by −ξi , it is then equivalent to assume instead of (14) that
d
drξi · φ(r) + C ≥ 0 in D′(IRN ) for i = 1, . . . , N − 1.
Now we notice that for η ∈ IRN , u is a bounded g.e.s. of (CP) corresponding to
φ, f, g if and only if u(t, x) = u(t, x − tη) is a bounded g.e.s. of (CP) corresponding to
φ(r) = φ(r)+ rη , f(x) = f(x) , and g(t, x) = g(t, x− tη) . Then, according to Corollary 1,
the conclusion of Theorem 3 holds for φ if and only if it holds for the flux function φ(r)+rη .
Choosing η ∈ IRN such that η · ξi > C for i = 1, . . . , N −1 , which is always possible since
the vectors are linearly independent, we may assume that
r ∈ IR 7→ ξi · φ(r) ∈ IR is an increasing homeomorphism for i = 1, . . . , N − 1, (15)
a slightly strengthened version of (3).
Under the assumption (15), we prove the result by induction in the dimension N . The
result is true for N = 1 (see [B72]). Assuming that it is true for N − 1 , we prove it for
N ≥ 2 . Changing coordinates, we may assume from (15) that φ(r) =(φ1(r), . . . , φN(r)
)
with φi(·) increasing homeomorphism from IR to IR for i = 1, . . . , N − 1 . We shall
prove that the equation (E) has a unique bounded g.e.s. for any f ∈ L1(IRN) ∩ L∞(IRN) .
According to Corollary 1, this will end the proof of the theorem.
So let f ∈ L1(IRN)∩L∞(IRN) and u be a bounded g.e.s. of (E); one has u ∈ L1(IRN)
(see Corollary 2.1 in [BK96]). Set x = (x1, x′) with x′ = (x2, . . . , xN) , w(x1, x
′) =
φ1(u(x1, x′)) , β = φ−1
1 , ψ(r) =(φ2(β(r)), . . . , φN(β(r))
). Suppose that w(x1 + t, ·) →
w(x1, ·) in L1loc(IR
N−1) as t → 0 for some x1 ∈ IR ; then for every T > 0 the function
v : (t, x′) ∈ Q′ = (0, T )× IRN−1 7→ w(x1 + t, x′) is a bounded g.e.s. of the Cauchy problem
∂v
∂t+ divx′ ψ(v) = g on Q′, v(0, ·) = v0(·) on IRN−1, (16)
where v0(x′) = w(x1, x
′) and g(t, x′) = f(x1+t, x′)−β(w(x1+t, x′)) ; g ∈ L1(Q′)∩L∞(Q′)
since f and β(w) = u are in L1(IRN ) ∩ L∞(IRN) .
According to Remark 2, it suffices to prove the uniqueness of a bounded g.e.s. of (E)
corresponding to compactly supported f . So assume supp f ⊂ H = x1 ≥ α0 and suppose
there exist u, u two bounded g.e.s. of (E). By Lemma 4, supp u ⊂ H , supp u ⊂ H . Take
x1 = α < α0 ; consider v(t, x′) = φ1(u(t+α, x
′)) , v(t, x′) = φ1(u(t+α, x′)) . The functions
94 Scalar Conservation Law with Continuous Flux Function
v, v are bounded g.e.s. of (16) corresponding to(v0(·) ≡ 0 , g(t, ·) = f(t+α, ·)−β(v(t, ·))
)
and(v0(·) ≡ 0 , g(t, ·) = f(t+ α, ·)− β(v(t, ·))
), respectively.
By the inductive assumption the Cauchy problem (16) has a unique bounded g.e.s., which
is in L1(IRN−1) for a.a. t ∈ (0, T ) ; as in the proof of Corollary 1, it follows that the integral
inequality holds:
d
dt
∫|v(t)− v(t)| ≤
[v(t)− v(t), g(t)− g(t)
]= −
∫|β(v(t))− β(v(t))| ≤ 0.
Hence
∫|v(t)− v(t)| ≤
∫|v0 − v0| = 0 , so that v = v a.e. in Q′ . Thus u = u a.e. in
H , which proves the Theorem. ⋄In [B72] it has been proved that for any f ∈ L1(IRN) ∩ L∞(IRN ) there is uniqueness of
u ∈ L1(IRN)∩L∞(IRN) g.e.s. of (E) under the isotropic assumption limr→0
‖φ(r)‖
r1−1/N = 0 . In the
next theorem we shall prove the uniqueness under the anisotropic assumption introduced in
[KP90] and [BK96].
Theorem 4 Let c ∈ IR and ω1, . . . , ωN be moduli of continuity, i.e., increasing sub-additive
continuous functions from [0, δ] into [0,+∞) , δ > 0 , with ωi(0) = 0 , i = 1, . . . , N ;
assume that
lim infr→0
1
rN−1
N∏
i=1
ωi(r) <∞. (17)
Assume that there exist orthonormal vectors ξ1, . . . , ξN such that |ξi·φ(c+r)−ξi·φ(c)| ≤ωi(|r|) for all r ∈ [−δ, δ] , i = 1, . . . , N . Then the two assertions of Corollary 1 hold†.
Proof: We may assume that c = 0 , φ(0) = 0 and φ = (φ1, . . . , φN) with |φi(r)| ≤ωi(|r|) for r ∈ [−δ, δ] , i = 1, . . . , N . Recalling Remark 2 and Corollary 1, we only need
to prove for f, u ∈ L1(IRN) ∩ L∞(IRN ) with u g.e.s. of (E) that
∫u =
∫f . Replacing
†In spite of the fact that this result underlines one more time (cf. [B72]) that Holder continuity of the flux
function at zero simplifies the issue of uniqueness in L1(IRN )∩L∞(IRN ) , Proposition 3 hereabove suggests
that the non-uniqueness, if there is any, can have no intrinsic relation with regularity properties of the flux
function. As we see, the appropriate Holder continuity of the flux function permits to pass to the limit directly
in the estimations of Lemma 3.1 in [BK96], and thus obtain the contraction property. But even in the case
one seems unable to deduce this property, there is uniqueness for a.a. translations φ(· − c)−φ(·) of the flux
function φ(·) , with c ∈ IR . This should be compared to the fact that one can construct functions that have
equally bad Holder continuity properties at all points of their domain; for exemple, one can use realisations of
a Wiener process. Thus we can suggest that, if the uniqueness in L1(IRN )∩L∞(IRN ) is determined by some
regularity property of the flux function, this must be a property that holds on the domain of any continuous
function φ(·) everywhere, except an at most countable set of points. The author ignores what could be the
nature of such property.
On the other hand, note that the assumption of Theorem 3 makes appeal to a global property of the flux
function, which also differs from the “pointwise regularity” point of view suggested by Theorem 4.
1.IV.3. Uniqueness results in L1(IRN) ∩ L∞(IRN) 95
ωi(r) , φi , f , and u by ωi(Mr)/M , φ(Mr)/M , f/M , and u/M , respectively, we
may assume that ‖u‖∞ ≤ δ .
Clearly, it suffices to show that for all µ > 0, R > 0 there exists a function ζ such that
0 ≤ ζ ≤ 1 on IRN , ζ(x) = 1 for all x ∈ [−R,R]N , and
∣∣∣∣∫
(u− f)ζ
∣∣∣∣ < µ; (18)
for this we follow the proof of Lemma 1.1 in [BK96].
For r > 0 set λi(r) = ωi(r)/r . If all λi are bounded, then φ is Lipschitz continuous
and the result is well known (see the Introduction). Without loss of generality we may assume
that limr→0
λi(r) = +∞ for i = 1, . . . , l and λi(r) ≤ λ for i = l+1, . . . , N with some
l ∈ 1, . . . , N . Since ωi are sub-additive and positive for r > 0 , ωi(r) ≥ λ0r for some
λ0 > 0 , so that it is equivalent to assume instead of (17) that lim infr→0
C(r) = C <∞ , where
C(r) = rλN−ll∏
i=1
λi(r) . Note that if l = 1 , then clearly C = 0 .
For all u bounded g.e.s. of (E), for all ζ ∈ D(IRN ) we have
∫uζ =
∫φ(u) ·Dζ +
∫fζ. (19)
Moreover, since f and u are bounded, (19) is also valid for ζ given by
ζ(x1, . . . , xN ) =N∏
i=1
exp
(−( |xi|Ri
− 1)+)
with arbitrary positive Ri . Take ζ corresponding to Ri = λi(ε)/η for i = 1, . . . , l and
Ri = λ/α for i = l+1, . . . , N ; positive numbers α, η, ε will be chosen later. We note
that 0 ≤ ζ ≤ 1 , ζ(x) ≡ 1 onN∏i=1
[−Ri, Ri] ,
∫ζ =
22N
ηlαN−lλN−l
l∏
i=1
λi(ε) , and |Diζ | =
ηζ
λi(ε)χ|xi|>Ri for i = 1, . . . , l , |Diζ | =
αζ
λχ|xi|>Ri for i= l+1, . . . , N .
From (19) we get
∣∣∣∣∫(u− f)ζ
∣∣∣∣ ≤N∑
i=1
∫ωi(|u|)|Diζ | .
Now, by the sub-additivity of ωi , for i = 1, . . . , l we have ωi(r) ≤ rωi(ε)/ε+ ωi(ε) =
rλi(ε)+ελi(ε) for all ε > 0 ; for i= l+1, . . . , N we have ωi(r) ≤ rλ . Hence by substituting
into the last estimate the expressions above for |Diζ | and
∫ζ , we get
∣∣∣∣∫
(u− f)ζ
∣∣∣∣ ≤l∑
i=1
η
∫
|xi|>Ri
|u|ζ +l∑
i=1
εη
∫
|xi|>Ri
ζ +N∑
i=l+1
α
∫
|xi|>Ri
|u|ζ ≤
≤ ηl∑
i=1
∫
|xi|>Ri
|u|+ l22N
ηl−1αN−l· ελN−l
l∏
i=1
λi(ε) + α(N − l)
∫|u|.
96 Scalar Conservation Law with Continuous Flux Function
Take µ > 0 , R > 0 . Choose α0 > 0 such that λ/α0 > R and α0 ·(N−l)‖u‖1 < µ/3 .
Choose η0 > 0 such that1
ηl−10
· l22N
αN−l0
· 2C < µ/6 ; note that if l = 1 then C = 0
and whatever η0 is good. Finally, since λi(ε) → ∞ as ε → 0 for i = 1, . . . , l and
u ∈ L1(IRN) , by definition of C there exists ε0 > 0 satisfying ε0λN−l
l∏
i=1
λi(ε0) < 2C +
µ
6· α
N−l0 ηl−1
0
l22Nsuch that Ri = λi(ε0)/η0 > R ,
l∑
i=1
∫
|xi|>Ri
|u| < µ
3η0. It follows that (18)
holds for ζ constructed with α0, η0, ε0 . ⋄
Remark 3 Introducing ξ1, . . . , ξN in the condition (17) is not superfluous. Indeed, take
N = 2 and let φ =(u , u/|u|2/3
)in some orthonormal basis ξ1, ξ2 ; here (17) holds.
Changing coordinates by rotation by any angle θ such that θ 6= πk/2, k ∈ ZZ , we see that
condition (17) fails in the new basis.
Appendix: proof of Lemma 2
We give here the complete proof of Lemma 2. More precisely, we shall prove the following
result:
Theorem 5 Let φ : IRN 7→ IR be a continuous function.
a) There exists a map G : L∞(IRN) 7→ L∞(IRN) satisfying:
(i) for any f ∈ L∞(IRN), u = Gf is a g.e.s. of (E);
(ii) G is a T-contraction for the L1-norm, i.e., for any f, f ∈ L∞(IRN)
∫ (Gf −Gf
)+≤∫ (
f − f)+.
b) Set X =(f, g) ; (f, g) satisfies (2)
; there exists a map U : X 7→ L∞(Q)∩
C([0, T ];L1loc(IR
N)) satisfying:
(i) for any (f, g) ∈ X , u = U(f, g) is a g.e.s. of (CP);
(ii) for any (f, g), (f , g) ∈ X , the T-contraction property holds:
supt∈[0,T ]
∫
IRN
(U(f, g)(t)− U(f , g)(t)
)+≤∫
IRN
(f − f
)++
∫∫
Q
(g − g
)+.
Proof of a): First, let f ∈ L1(IRN) ∩ L∞(IRN ) . For ε > 0 , take φε : IR → IRN
Lipschitz continuous functions such that φε converge to φ uniformly on compact sets in
IR , as ε→ 0 . It is well-known that there exists a unique solution uε to the equation
uε + divx φε(uε) = ε∆x u
ε + f on IRN ;
moreover, the map Gε : f ∈ L1(IRN) ∩ L∞(IRN) 7→ uε ∈ L1(IRN ) ∩ L∞(IRN) is a T-
contraction for the L1 -norm, the maximum principle ( ‖uε‖∞ ≤ ‖f‖∞ ) holds and there is
1.IV.4. Appendix 97
translation invariance in x . Thus the family Gεfε>0 is relatively compact in L1loc(IR
N) .
Take a countable L1 -dense set M in L1(IRN) ∩ L∞(IRN) ; by the diagonal process, there
exist εn → 0 such that Gεnf → u =: G0f in L1loc(IR
N) for all f ∈ M . It is clear
that u is a g.e.s. of (E) and G0 : M 7→ L1(IRN) ∩ L∞(IRN) is a T-contraction for the
L1 -norm. Thus G0 can be extended to the whole of L1(IRN) ∩ L∞(IRN) so that G0 is a
T-contraction for the L1 -norm, G0f is a g.e.s. of (E) and the maximum principle holds.
Now for the general case f ∈ L∞(IRN) , set fn,m = f+χ|x|≤n−f−χ|x|≤m ∈ L1(IRN)∩L∞(IRN ) . As n → ∞ , G0fn,m ↑ um ∈ L∞(IRN) ; further, as m → ∞ , um ↓ u =:
Gf . It is clear that u is a bounded g.e.s. of (E); by the Fatou Lemma, it follows that∫ (Gf − Gf
)+≤ lim inf
n→∞,m→∞
∫ (fn,m − fn,m
)+for f, f ∈ L∞(IRN ) . It is easy to check
that limm→∞
limn→∞
∫ (fn,m − fn,m
)+=
∫ (f − f
)+∈ [0,+∞] , so that (ii) also holds. ⋄
Proof of b): First, set X0 =[L1(IRN)∩L∞(IRN)
]×[L1(Q)∩L∞(Q)
]and let (f, g) ∈
X0 . For ε > 0 , take φε as in the proof of a); there exists a unique solution uε to the
Cauchy problem
∂uε
∂t+ divx φ
ε(uε) = ε∆x uε + g on Q
uε(0, ·) = f on IRN ;
moreover, the map Uε : (f, g) ∈ X0 7→ uε ∈ L1(Q) ∩ L∞(Q) ∩ C([0, T ];L1loc(IR
N)) satisfies
the maximum principle ( ‖uε‖∞ ≤ ‖f‖∞ +
∫ T
0
‖g(τ, ·)‖∞dτ ), the T-contraction property
holds , and there is translation invariance in x . Hence there exists a modulus of continuity
ωf,g such that∫ ∣∣∣Uε(f, g)(t, x+∆x)− Uε(f, g)(t, x)
∣∣∣dx ≤ ωf,g(∆x)
uniformly in ε > 0 and t ∈ [0, T ] . By Theorem 2 in [K69a], it follows that for any compact
set K ⊂ IRN
∫
K
∣∣∣Uε(f, g)(t+∆t, x)− Uε(f, g)(t, x)∣∣∣dx ≤ ωf,g,K(∆t)
uniformly in ε > 0 and t ∈ [0, T ] , where ωf,g,K is a modulus of continuity. Take a
countable set M dense in X0 for the L1(IRN)×L1(Q) -topology. By the diagonal process,
there exist εn → 0 such that Uεn(f, g) → u =: U0(f, g) in L1loc(Q) for all (f, g) ∈ M ;
u is a g.e.s. of (CP), and the maximum principle and the T-contraction property hold for
U0 : M 7→ L1(Q) ∩ L∞(Q) ∩ C([0, T ];L1loc(IR
N)) . Thus U0 can be extended to the whole
of X0 , so that U0(f, g) is a g.e.s. of (CP), the T-contraction property holds and there is
translation invariance in x and the maximum principle holds.
Now for the general case (f, g) ∈ X , set fn,m = f+χ|x|≤n − f−χ|x|≤m and gn,m =
minn, g+χ|x|≤n−minm, g−χ|x|≤m , so that we have (fn,m, gn,m) ∈ X0 . As n→ ∞ ,
98 Scalar Conservation Law with Continuous Flux Function
U0(fn,m, gn,m) ↑ um ∈ L∞(Q) ; further, as m → ∞ , um ↓ u =: U(f, g) ∈ L∞(Q) . By the
Fatou Lemma, it follows that (ii) holds.
We now show that u = U(f, g) is in C([0, T ];L1loc(IR
N )) and a g.e.s. of (CP). Indeed,
there exists an increasing sequence (ni, mi)i∈IN in IN2 such that ui = U0(fi, gi) → u
in L1loc(Q) as i → ∞ , where fi = fni,mi
, gi = gni,mi. By Lemma 3.1 in [BK96] and
translation invariance, for ζ ∈ D(IRN )
supt∈[0,T ]
∫ ∣∣∣ui(x+∆x, t)−ui(x, t)∣∣∣ ζ(x) dx ≤
∫ ∣∣∣fi(x+∆x)−fi(x)∣∣∣ ζ(x) dx+
+
∫ T
0
∫ ∣∣∣gi(s, x+∆x)−gi(s, x)∣∣∣ ζ(x) dxds+
∫ T
0
∫ ∣∣∣φ(ui(s, x+∆x))−φ(ui(s, x))∣∣∣∣∣∣Dζ(x)
∣∣∣ dxds.
The last term tends to
∫ T
0
∫ ∣∣∣φ(u(s, x+∆x))−φ(u(s, x))∣∣∣∣∣∣Dζ(x)
∣∣∣ dxds as i→ ∞ , therefore
for any compact set K ⊂ IRN
∫
K
∣∣∣ui(t, x+∆x)− ui(t, x)∣∣∣dx ≤ ωf,g,K(∆x)
uniformly in i ∈ IN and t ∈ [0, T ] , where ωf,g,K is a modulus of continuity. Hence, again by
Theorem 2 in [K69a], the family ui(t, ·)i∈IN is equicontinuous from [0, T ] to L1loc(IR
N) .
Thus u ∈ C([0, T ];L1loc(IR
N)) and u(0, ·) = limi→∞
fi = f , so that u is a bounded g.e.s. of
(CP). ⋄
Part 2
Weak Solutions
for Elliptic-Parabolic Systems
CHAPTER 2.I
Elliptic-Parabolic Problems: Existence
and Continuity with Respect to the Data
of Weak Solutions†
Introduction
Let Ω be a bounded domain of IRd with Lipschitz boundary ∂Ω . For i = 1, . . . , N , let
Γ0,i be a closed set in ∂Ω and Γ1,i = ∂Ω\Γ0,i . We consider initial boundary value problems
for elliptic-parabolic systems of the form
b(·, v)t = div a(·, v, Dv) + f(·, v) on Q = (0, T )× Ω
ai(·, v, Dv) · ν = gi(·, v) on Σ1,i = (0, T )× Γ1,i for i = 1, . . . , N
vi = hi on Σ0,i = (0, T )× Γ0,i for i = 1, . . . , N
b(·, v)(0, ·) = u0 on Ω,
(P)
where b : Q× IRN 7→ IRN , a = (a1, . . . , aN) : Q× IRN × (IRd)N 7→ (IRd)N , f : Q× IRN 7→IRN , g = (g1, . . . , gN) : Σ × IRN 7→ IRN , h = (h1, . . . , hN) : Σ 7→ IRN , u0 : Ω 7→ IRN ,
and ν is the external unit normal vector to Σ = (0, T )× ∂Ω . We denote by “ · ” the scalar
product in IRN , and by “ : ” the scalar product in (IRd)N . Except in Section 3.3, we assume
that |Γ0| =∏N
i=1 |Γ0,i| > 0 .
The assumptions on the data will be made precise in Section 1. In the particular case
b(t, x, z) = b(z) , a(t, x, z, ξ) = a(z, ξ) , f ≡ 0 , g ≡ 0 , h ≡ 0 , the assumptions reduce to
the three following conditions on b, a, u0 :
b = ∂Φ, with Φ : IRN 7→ IR convex differentiable ,Φ(0) = 0; (1)
†This chapter is being prepared upon publication [BAB]
102 Continuous Dependence for Elliptic-Parabolic Problems
as usual, we set B(z) =
∫ 1
0
(b(z) − b(σz)
)· z dσ = b(z) · z − Φ(z) ≥ 0 for any z ∈ IRN ;
a is continuous, monotone in ξ, and satisfies Leray-Lions type conditions :
there exist c > 0, C ≥ 0 and a “sublinear” function L : IR+ 7→ IR+
(i.e., a function with limr→∞
L(r)/r = 0) such that
a(z, ξ) : ξ ≥ c|ξ|p −L(|z|p)− C(1 +B(z)
),
|a(z, ξ)|p′ ≤ L(B(z) + |z|p
)+ C
(1 + |ξ|p
),
(2)
with 1 < p <∞ , p′ = p/(p− 1) ;
u0 ∈ L1(Ω; IRN ) with Ψ(u0) ∈ L1(Ω), (3)
where Ψ is the Legendre conjugate of Φ defined by Ψ(z∗) = supz∈ IRN
(z · z∗ − Φ(z)
)∈
[0,+∞] for any z∗ ∈ IRN . Note that one has B(z) = Ψ(b(z)) for all z ∈ IRN . In this
chapter, adding the structure conditiona(z, ξ) = a(b(z), ξ) for all z ∈ IRN , ξ ∈ (IRd)N ,
where a : R(b(·))× (IRd)N 7→ (IRd)N is Caratheodory,(4)
we prove existence of a weak solution (in the variational sense) to (P ) .
Such result has already been proved by Alt-Luckhaus in [AL83], while under more restrictive
assumptions. Since then, many similar results have been obtained (e.g., cf. [Kac90, DiDT94,
FKac95, BW96, Bou97]), in particular in the case of time-space dependent elliptic part and
space dependent parabolic part. Equations with the parabolic part of the form c(t, x, v) vt
have been studied in [Pl98, Pl00] by a different approach; existence results for some of equa-
tions of this form can be derived from the ours, upon introducing b(t, x, z) =
∫ z
0
c(t, x, ζ) dζ .
The aim and the interest of this work is, first, in proving the continuous dependence of
weak solutions on the data in (P ) ; existence is an easy corollary of this result. Secondly,
we use arguments that permit to make less restrictive assumptions (in particular, all data are
(t, x) - dependent) and, at the same time, clarify the essence of the proof.
The continuity theorem is presented in Section 1 (Theorem 1); in the reduced case con-
sidered above it reads as follows:
Theorem 0 Let(bn, an, u0,n
)be a sequence of data satisfying the assumptions (1)-(3) with
c, C,L(·) independent of n ∈ IN and corresponding Bn,Ψn . Assume that bn(·) → b(·) in
C(IRN ; IRN ) †, for all ξ ∈ (IRd)N an(·, ξ) → a(·, ξ) in C(IRN ; (IRd)N) , and u0,n → u0 a.e
on Ω with Ψn(u0,n) → Ψ(u0) in L1(Ω) . For n = 1, 2, . . . let vn be a weak solution of
the corresponding problem (Pn) with f ≡ 0 , g ≡ 0 , h ≡ 0 , and where |Γ0| > 0 . Then
†I.e., for all compact subset K of IRN there is uniform convergence for z ∈ K
2.I.1. Assumptions and results 103
(i) The sequence vn is bounded in Lp(0, T ;W 1,p(Ω; IRN)) and the sequence bn(vn)is relatively compact in L1(Q; IRN) .
(ii) Any weak limit point of vn in Lp(0, T ;W 1,p(Ω; IRN)) is a weak solution of (P ) ,
provided (P ) satisfies (4).
A precise definition of a weak solution is given in Section 1. Note that, while not assuming
the strict monotonicity of b in z , we have to impose the structure condition (4) , which
is trivial for b strictly monotone. However, this condition is not intrinsically related to the
existence of weak solution to (P ) (cf. [BW99] and Section 3.2).
The proof of the continuity theorem in general setting is given in Section 2 and includes
three essential arguments. First, we establish a time-dependent version of the chain rule
Lemma 1.5 from [AL83] and apply it to get a priori estimates which imply compactness in
x of vn . Secondly, using a general compactness lemma due to S.N.Kruzhkov ([K69a]) we
deduce the compactness of bn(vn) in (t, x) . Finally, under additional structure conditions,
the Minty-Browder argument is used for passage to the limit in the equation (Pn) . The proofs
of the appropriate versions of this three results (Lemma 1, Lemma 6 and Lemma 7) are given
in the Appendix.
In Section 3 we give some remarks and further existence results. More specifically, we
treat the case of inhomogeneous Dirichlet data, discuss structure conditions of type (4), give
one direct extension to the case |Γ0| = 0 , and indicate possible applications to proving
convergence of approximate methods.
1 Assumptions and results
In this section and Section 2 we state the results for the problem (P ) with h ≡ 0 and
|Γ0| > 0 . See Section 3 for some comments on the general case.
Let Ω be a bounded set in IRd with Lipschitz boundary ∂Ω . For i = 1, . . . , N , let
Γ0,i be a closed set in ∂Ω and Γ1,i = ∂Ω\Γ0,i . Let 1 < p <∞ , N be a positive integer.
Set V =w ∈ W 1,p(Ω; IRN) : wi = 0 a.e. on Γ0,i, i = 1, . . . , N
. Let T > 0 , and set
Q = (0, T )× Ω , Σ = (0, T )× ∂Ω .
Our assumptions include a collection of hypotheses that impose restrictions on the growth
of the coefficients ( (H2), (H3), (H4), (H6), (H7), (H8), (H13) below). We will say that
104 Continuous Dependence for Elliptic-Parabolic Problems
this collection of hypothesis is satisfied if there exist
c = const > 0,
C = const ≥ 0,
K0 : [0, T ]× Ω× IR+ 7→ IR+ such that
K0(·, r) ∈ L1(Q) and K0(0, ·, r) ∈ L1(Ω) for all r ∈ IR+,
K1 : (0, T ) 7→ IR+, K1 ∈ L1(0, T ),
K2 : Q 7→ IR+, K2 ∈ L1(Q),
K3 : Σ 7→ IR+, K3 ∈ L1(Σ),
(5)
for all ε > 0 there exist Kε , Kε1 , K
ε2 and Kε
3 such that
Kε = const ≥ 0,
Kε1 : (0, T ) 7→ IR+, Kε
1 ∈ L1(0, T ),
Kε2 : Q 7→ IR+, Kε
2 ∈ L1(Q),
Kε3 : Σ 7→ IR+, Kε
3 ∈ L1(Σ),
(6)
and if the corresponding inequalities holds.
Let b : [0, T ]× Ω× IRN 7→ IRN satisfy
b(t, x, z) = ∂Φ(t, x, ·)(z), where Φ : Q× IRN 7→ IR
is convex differentiable in z and Φ(t, x, 0) = 0 for all t ∈ [0, T ], x ∈ Ω;(H1)
|b(t, x, z)| ≤ K0(t, x, |z|) for all z ∈ IRN , t ∈ [0, T ] and a.a. x ∈ Ω. (H2)
Set B(t, x, z) =
∫ 1
0
(b(t, x, z) − b(t, x, σz)
)· z dσ , and Ψ(t, x, z∗) = Φ(t, x, ·)∗(z∗) =
supz∈ IRN
(z ·z∗−Φ(t, x, z)
)for any z∗ ∈ IRN . One has B(t, x, z) = b(t, x, z) ·z−Φ(t, x, z) =
Ψ(t, x, b(t, x, z)
).
Note that one has (e.g., cf. Remark 1.2 in [AL83]) for all δ > 0 ,
|b(t, x, z)| ≤ δB(t, x, z) + sup|ζ|≤1/δ
|b(t, x, δ)| ≤ δB(t, x, z) +K0(t, x, 1/δ). (7)
In addition to (H1), (H2) we assume that Φ is absolutely continuous in t on [0, T ]
for all z ∈ IRN and a.a. x ∈ Ω , and that there exists Φt : Q × IRN 7→ IR Caratheodory
such thatd
dtΦ(·, x, z)(t) = Φt(t, x, z) for all z ∈ IRN and a.a. (t, x) ∈ Q . Moreover, we
require that for a.a. x ∈ Ω , a.a. t, s ∈ (0, T ) and all z ∈ IRN the function Φt satisfy
|Φt(t, x, z)| ≤ ε(K1(t)B(s, x, z) + |z|p
)+Kε
2(t, x). (H3)
2.I.1. Assumptions and results 105
It follows that Φ, b, B are measurable in x for all t ∈ [0, T ] and z ∈ IRN , and continuous
in (t, z) for a.a. x ∈ Ω .
Let f : Q× IRN 7→ IRN be Caratheodory and assume for a.a. (t, x) ∈ Q
|f(t, x, z)|p′ ≤ ε(K1(t)B(t, x, z) + |z|p
)+Kε
2(t, x). (H4)
Let a : Q× (IRN × (IRd)N) 7→ (IRd)N be Caratheodory and assume for a.a. (t, x) ∈ Q(a(t, x, z, ξ)−a(t, x, z, ξ)
): (ξ− ξ) ≥ 0 for all z ∈ IRN , ξ, ξ ∈ (IRd)N , (H5)
a(t, x, z, ξ) : ξ − f(t, x, z) · z + Φt(t, x, z) ≥ c|ξ|p − ε |z|p−−Kε
1(t)B(t, x, z)−Kε2(t, x),
(H6)
|a(t, x, z, ξ)|p′ ≤ ε(K1(t)B(t, x, z) + |z|p
)+Kε
2(t, x) + C|ξ|p. (H7)
Let gi : (0, T )×Γ1,i× IRN 7→ IRN be Caratheodory, i = 1, . . . , N . For convenience, for
all z ∈ IRN extend gi(·, z) by zero on (0, T )× Γ0,i , and assume for a.a. (t, x) ∈ Σ
|g(t, x, z)|p′ ≤ C|z|p +K3(t, x)
g(t, x, z) · z ≤ ε |z|p +Kε3(t, x).
(H8)
Remark 1 The growth restrictions in (H3), (H4), (H6)−(H8) are in fact of the same type
as those in (2). It is convenient to pass to the form chosen above, rather than majorate
|Φt(t, x, z)| , |f(t, x, z)|p′ , etc., by terms of the form L(t, x, B(t, x, z)) and L(t, x, |z|p) ,with “sublinear” functions L(t, x, ·) subject to additional restrictions on their dependence on
(t, x) .
A typical situation where, for instance, the hypothesis (H4) is satisfied, is when one has
|f(t, x, z)|p′ ≤ M(t, x)((B(t, x, z))1/κ + |z|p/κ
)+K2(t, x)
with κ > 1 , M ∈ Lκ′(Q) and K2 ∈ L1(Q) . An assumption of slightly different kind that
is also covered by (H4) is that
|f(t, x, z)|p′ ≤ N (t)L(B(t, x, z)) + L(|z|p) +K2(t, x)
with N ∈ L1(0, T ) and L(·) “sublinear”, independent of (t, x) . Other growth hypotheses
can be simplified in similar ways.
To shorten the notation, for all v : Q 7→ IRN we will denote the function b(t, x, v(t, x))
by b(v) , the function a(t, x, v(t, x), Dw(t, x)) by a(v,Dw) , the function Ψ(τ, x, w(x))
by Ψ(τ, w) and so on.
†It seems that a more natural condition would be (H3) written only for s = t ; see Remark 6 in the
Appendix for a discussion of this issue
106 Continuous Dependence for Elliptic-Parabolic Problems
Definition 1 For u0 ∈ L1(Ω; IRN) and h ∈ Lp(0, T ;W 1,p(Ω; IRN )) , a weak solution of
(P ) is a function v : Q 7→ IRN satisfying
(i) v ∈ h+ Lp(0, T ;V ) , B(v) ∈ L∞(0, T ;L1(Ω)) (whence b(v) ∈ L1(Q) by (7));
(ii) for all ζ ∈ Lp(0, T ;V ) with ζt ∈ L∞(Q) and ζ(T ) = 0 ,
∫∫
Q
b(v) · ζt +∫
Ω
u0(·) · ζ(0, ·) =∫∫
Q
a(v,Dv) : Dζ −∫∫
Q
f(v) · ζ −∫∫
Σ
g(v) · ζ.
Let denote by V ′ the dual space of V , and by < ·, · > the duality pairing between V ′ and
V . We remark that, according to (i) and the conditions (H4), (H7), (H8) , the condition
(ii) in the Definition 1 makes sense and can be rewritten, as in [AL83], under the equivalent
form:
there exists χ ∈ Lp′(0, T ;V ′) such that
(ii1)
∫∫
Q
b(v) · ζt +∫
Ω
u0(·) · ζ(0, ·) =∫ T
0
< χ, ζ >
for all ζ ∈ Lp(0, T ;V ) with ζt ∈ L∞(Q) and ζ(T ) = 0,
and
(ii2)
∫ T
0
< χ, ζ >=
∫∫
Q
a(v,Dv) : Dζ −∫∫
Q
f(v) · ζ −∫∫
Σ
g(v) · ζ
for all ζ ∈ Lp(0, T ;V ).
According to (ii1) , the distribution derivative of b(v) with respect to t , b(v)t = −χ , is in
Lp′(0, T ;V ′)) .
Following [AL83], in addition to u0 ∈ L1(Ω; IRN) we assume
Ψ(0, u0) ∈ L1(Ω). (H9)
One has the following result.
Lemma 1 Let b satisfy (H1), (H2), (H3) with corresponding functions B,Φ . Assume
that u0 ∈ L1(Ω; IRN) with Ψ(0, u0) ∈ L1(Ω) and v ∈ Lp(0, T ;V
)with B(v) ∈
L∞(0, T ;L1(Ω)) (whence b(v) ∈ L1(Q) ) be given such that (ii1) holds with some χ ∈Lp′(0, T ;V ′) . Then one has Φt(v) ∈ L1(Q) , and for a.a. t ∈ (0, T )
∫
Ω
B(v)(t) =
∫
Ω
Ψ(0, u0)−∫ t
0
< χ(τ), v(τ) > dτ −∫ t
0
∫
Ω
Φt(v)(τ) dτ. (8)
This is a time-dependent version of the chain rule lemma (cf. [Bam77, AL83, Ot96, CaW99]),
which is crucial in this framework. We give a proof of it in the Appendix.
2.I.1. Assumptions and results 107
Remark 2 Under the hypothesis we take, we avoid much unnecessary technicalities by re-
quiring into Definition 1 and Lemma 1 that B(v) ∈ L∞(0, T ;L1(Ω)) (which was one of the
claims in the corresponding lemma in [AL83]). Still this property, together with (ii1) and
(ii2) , could be deduced from the assumption that b(v) ∈ L1(Q) and Definition 1 (ii) . We
also take h ≡ 0 in the statement of Lemma 1, hoping to get the general case by translation
by h (cf. Section 3.1).
In order to state the main result of this chapter, we consider a sequence of problems (Pn) ,
n ∈ IN , with data(bn, an, fn, gn, u0,n
)satisfying
(bn, an, fn, gn
)verify (H1)− (H8) with c,K0, K1, K
ε, Kε2 , K3
independent of n and corresponding functions Bn,Φn,t,Ψn;
Ψn(0, u0,n) is bounded in L1(Ω).
(H)
We will say that a sequence of problems (Pn) converge to the problem (P ) with data(b, a, f, g, u0
), if:
(bn(t, x, ·),Φn,t(t, x, ·), an(t, x, ·, ξ), fn(t, x, ·)
)−→
−→(b(t, x, ·),Φt(t, x, ·), a(t, x, ·, ξ), f(t, x, ·)
)
in C(IRN ; IRN × (IRd)N × IRN
)for a.a. (t, x) ∈ Q, for all ξ ∈ (IRd)N ;
gn(t, x, ·) → g(t, x, ·) in C(IRN ; IRN) for a.a. (t, x) ∈ Σ;
u0,n → u0 a.e. on Ω with Ψn(0, u0,n) → Ψ(0, u0) in L1(Ω)
(9)
Theorem 1 Let |Γ0| > 0 and the Dirichlet part of the boundary data be homogeneous, i.e.,
h ≡ 0, (H10)
and(bn, an, fn, gn, u0,n
)be a sequence of data satisfying (H) . Assume that (Pn) converge
to (P) in the sense (9). For n = 1, 2, . . . let vn be a weak solution of the corresponding
problem (Pn) . Then
(i) The sequence vn is bounded in Lp(0, T ;W 1,p(Ω; IRN)) and the sequence bn(vn)is relatively compact in L1(Q; IRN) .
(ii) Any weak limit point of vn in Lp(0, T ;W 1,p(Ω; IRN)) is a weak solution of (P ) ,
provided (P ) satisfies the structure conditions (H11)− (H13) †:f(t, x, z) = f(t, x, b(t, x, z)) for all z ∈ IRN and a.a. (t, x) ∈ Q,
where f : (t, x, β) ∈ Q× IRN | β ∈ R(b(t, x, ·)) 7→ IRN is Caratheodory;(H11)
†By R(b(t, x, ·)) we denote the image of IRN by b(t, x, ·)
108 Continuous Dependence for Elliptic-Parabolic Problems
a(t, x, z, ξ) = a(t, x, b(t, x, z), ξ) for all z ∈ IRN , ξ ∈ (IRd)N and a.a. (t, x) ∈ Q,
where a : (t, x, β, ξ) ∈ Q× (IRN × (IRd)N) | β ∈ R(b(t, x, ·)) 7→ (IRd)N
is Caratheodory;
(H12)
Φt(t, x, z) = φ(t, x, b(t, x, z)) + z · ϕ(t, x, b(t, x, z)) for all z ∈ IRNand a.a. (t, x) ∈ Q,
where (φ, ϕ) : (t, x, β) ∈ Q× IRN | β ∈ R(b(t, x, ·)) 7→ IR× IRN
is Caratheodory and‡ |ϕ(t, x, b(t, x, z))|p′ ≤ K1(t)Ψ(t, x, b(t, x, z)) + C|z|p +K2(t, x)
for a.a. (t, x) ∈ Q and all z ∈ IRN .
(H13)
Remark 3 In the convergence part, we will manage with the dependence of coefficients on
(t, x) by considering them as mappings from Q to C(IRN ; IRN ) or C(IRN×(IRd)N ; (IRd)N)
and repeatedly applying the Egorov and Lusin theorems. It is possible, since a Caratheodory
function Y1× Y2 7→ Y3 , where Yi ⊂ IRDi , i = 1, 2, 3 , is measurable considered as mapping
from Y1 to C(Y2; Y3) . Indeed, it is weakly measurable since it is measurable in y1 ∈ Y1 for
all fixed y2 ∈ Y2 ; hence it is strongly measurable (e.g., cf. [BbkiXIII],Chap.IV,S5,Prop.10).
Corollary 1 Let the data in (P ) satisfy (H1)− (H13) , and Ψ(u0) ∈ L1(Q) . Then there
exists a weak solution to (P ) with h = 0 .
Proof of Corollary 1: Let approximate (P ) in the sense (H) ,(9) by a sequence of
problems (Pn) with bn bilipschitz in z , and with data and coefficients regular in (t, x) .
Note that we have to assure that Ψn(0, u0,n) → Ψ(0, u0) in L1(Ω) . This is done by
choosing for all n ∈ IN some m ∈ IN and a measurable function u0,n such that u0,n
is a value within the image of [−m,m]N by b(0, x, ·) , and ‖Ψn(0, u0,n) − Ψ(0, u0)‖ ≤1/n . In turn, this is possible since Bn(0, x, ·) are dominated on [−m,m]N by the function√nmK0(0, x,
√nm) ∈ L1(Ω) uniformly in n , since the assumption (H2) holds uniformly
in n .
To prove the existence for (Pn) itself, we use the Galerkin approximations in the way it is
done in [LJLL65, JLL] and [ALpr]. The global in time existence of Galerkin approximations for
(Pn) follows from an a priori L∞(0, T ;L2(Ω)) estimate, obtained as in [ALpr]. It is at this
level that we also find that B(v) ∈ L∞(0, T ;L1(Ω)) , which we have required in Definition 1.
The convergence of Galerkin approximations for (Pn) follows as in Theorem 1 and yields
existence of a weak solution vn .
Applying Theorem 1 to the constructed sequence vn , we get existence for (P ) . ⋄‡This hypothesis actually seems amount to nothing more than this inequality. Moreover, a natural candidate
for ϕ is bt , and this inequality, with (H3) and (H4) taken into account, means that it could be considered
as a part of the second member in (P ) . See Lemma 3 for a discussion of the condition (H13) in the scalar
case.
2.I.2. Proof of the continuity theorem 109
2 Proof of the continuity theorem
In order to prove Theorem 1, we start with the following lemma:
Lemma 2 Assume (P ) satisfy (H1), (H2), (H3), (H6), (H8) and Ψ(0, u0) ∈ L1(Ω) ; let
v is a weak solution of (P ) . Then
(i) supτ∈[0,T ]
∫
Ω
B(v)(τ, ·) + ‖v‖Lp(0,T ;W 1,p(Ω; IRN )) ≤M
(ii) there exists a function ω ∈ C(IR+; IR+) , ω(0) = 0 such that for all E ⊂ Q
∫∫
E
|b(v)| ≤ ω(|E|);
here M and ω(·) are determined solely by ‖Ψ(0, u0)‖L1(Ω) , c,K0, K1, K2 in (5) and the
dependence of Kε, Kε2 in (6) on ε .
Proof of Lemma 2: Take vχ[0,t)×Ω for the test function in (ii2) of Definition 1. Applying
Lemma 1, on account of (H3), (H6), (H8) we get for a.a. t ∈ (0, T )
∫
Ω
B(v)(t) −∫
Ω
Ψ(0, u0) + c
∫ t
0
∫
Ω
|Dv|p ≤ ε(∫ t
0
∫
Ω
|v|p +
∫ t
0
∫
∂Ω
|v|p)+
+
∫ t
0
Kε1(τ)
∫
Ω
B(v)(τ) dτ +
∫ t
0
∫
Ω
Kε2 +
∫ t
0
∫
∂Ω
Kε3
with ε > 0 and the corresponding Kε1 , K
ε2, K
ε3 . Hence, using the imbedding of W 1,p(Ω)
into Lp(∂Ω) and the Poincare inequality, we get for all ε sufficiently small
∫
Ω
B(v)(t) +c
2‖v‖Lp(0,T ;W 1,p(Ω;IRN )) ≤ ‖Ψ(u0)‖L1(Ω) +
+
∫ t
0
Kε1(τ)
∫
Ω
B(v)(τ) dτ + ‖Kε2‖L1(Q) + ‖Kε
3‖L1(Σ).
Thus (i) follows from the Gronwall inequality.
Consequently, (ii) follows readily from (7); it suffices to take
ω(r) = minδ>0
(Mδ + sup
E⊂Q,|E|≤r
∫∫
E
K0(·, 1/δ)). ⋄
Now let prove the compactness part of the continuity theorem.
Proof of (i) in Theorem 1: By Lemma 2, ‖vn‖Lp(0,T ;W 1,p(Ω; IRN )) are bounded;
moreover, un = bn(vn) are equiintegrable on Q .
Let prove the “compactness in x ” of un in L1(Q) (i.e. the property (10) below); its
compactness in (x, t) in L1(Q) will follow from Lemma 6 (see the Appendix and [K69a]).
Indeed, un = bn(vn) satisfies the evolution equation ∂/∂t un = Fn in D′(Q) , with Fn
110 Continuous Dependence for Elliptic-Parabolic Problems
bounded in L1(0, T ;W−1,1(Ω)) by virtue of (H4), (H7), (H8) and (i) of Lemma 2. More-
over, un is bounded in L1(Q) by (ii) of Lemma 2. So we only need to show that for all
compact set K ⊂ Ω , for all h ∈ [0, dist(K, ∂Ω)/2] one has
In = sup|∆x|≤h
∫ T
0
∫
K
|un(t, x+∆x)− un(t, x)∣∣∣ dxdt ≤ ωK(h) (10)
with some function ωK ∈ C(IR+; IR+) with ωK(0) = 0 , ωK independent of n .
Fix α > 0 . First, by Remark 3 and the Lusin and Egorov theorems there exists an
open set Qα ⊂ Q , |Qα| < α such that bn → b in C(Q \ Qα, C(IRN ; IRN )) = C((Q \
Qα) × IRN ; IRN) . Thus it follows by the Arzela-Ascoli theorem that there exists a function
ωα,M ∈ C((IR+)3; IR+) with ωα,M(0, 0, 0) = 0 such that for all (t, x), (t′, x′) ∈ Q \Qα , all
z, z′ ∈ [−M,M ]N one has |bn(t, x, z)− bn(t′, x′, z′)| ≤ ωα,M
(|t− t′|, |x− x′|, |z − z′|
).
Second, take M = M(α) = 1/α supn ‖vn‖L1(Q) ; by the Chebyshev inequality, for each
n ∈ IN there exists another set Qα,n ⊂ Q , |Qα,n| < α such that |vn| ≤ M(α) a.e. on
Q \Qα,n .
Now we can estimate In in (10) by integrating separately over the set Qα,n(∆x) =
(t, x) ∈ (0, T )×K : (t, x) ∈ Qα ∪Qα,n or (t, x+∆x) ∈ Qα ∪Qα,n , with |Qα,n(∆x)| <4α , and the complementary set Q′
α,n(∆x) = ((0, T )×K) \Qα,n(∆x) . By the concavity of
ωα,M(0, |∆x|, ·
), we get
In ≤ ω(4α) +
∫∫
Q′α,n(∆x)
|bn(t, x+∆x, vn(t, x+∆x))− bn(t, x, vn(t, x))| dxdt ≤
≤ ω(4α) + +
∫∫
Q
ωα,M(0, |∆x|, |vn(t, x+∆x)− vn(t, x)|
)≤ ω(4α) +
+ |Q|ωα,M(0, |∆x|, 1
|Q|
∫∫
Q
|vn(t, x+∆x)− vn(t, x)|)
≤
≤ ω(4α) + |Q|ωα,M(0, h, sup
n
‖Dvn‖L1(Q)
|Q|h).
Minimizing the right-hand side in α > 0 , we get a function ωK with the desired properties.
⋄
Finally, let prove the convergence part of the continuity theorem.
Proof of (ii) in Theorem 1: By compactness, choose a subsequence (which we still
denote by vn ) such that vn v in Lp(0, T ;V ) , vn → v in Lp(Σ; IRN) and a.e., and
bn(vn) → u in L1(Q; IRN ) and a.e. on Q .
It follows that u = b(v) , by the argument introduced in [BrSt73]. More precisely, by
Remark 3, the Egorov theorem and the Chebyshev inequality, for all α > 0 there exists an
open set Qα ⊂ Q such that |Qα| < α , bn → b in C(Q \ Qα;C(IRN)) , bn(vn) → u in
L∞(Q \Qα) , and v is bounded on Q \Qα . Therefore for all η, ζ ∈ L∞(Q \Qα) we have
bn(η) → b(η) in L∞(Q \Qα) ; in addition, b(v + λζ) → b(v) in L∞(Q \Qα) as λ→ 0 .
2.I.2. Proof of the continuity theorem 111
Thus, by the monotonicity of bn ,∫∫
Q\Qα
u(v − η) = lim
∫∫
Q\Qα
bn(vn)(vn − η) ≥
≥ lim
∫∫
Q\Qα
bn(η)(vn − η) =
∫∫
Q\Qα
b(η)(v − η).
Taking η = v + λζ with arbitrary ζ ∈ L∞(Q \ Qα) , letting λ decrease to 0 and then
increase to 0 , we conclude that u = b(v) in L1(Q \ Qα) . Letting α go to 0 , we get
u = b(v) a.e. on Q .
It follows that bn(vn)t → b(v)t in Lp′(0, T ;V ′) ; indeed, note that bn(vn)t are uni-
formly bounded in this latter space by (ii2) of Definition 1, Lemma 2 and hypotheses
(H4), (H7), (H8) .
The initial condition (ii1) is therefore satisfied at the limit; indeed, (9) implies that
u0,n → u0 in L1(Ω; IRN) . Note also that, by the Fatou lemma, B(v) ∈ L∞(0, T ;L1(Ω)) ,
which will permit to apply Lemma 1 to the function v .
Let prove that (ii2) holds as well.
Start by showing that one has gn(vn) → g(v) and fn(vn) → f(v) in Lp′(Σ; IRN) and
Lp′(Q; IRN) , respectively. The former follows readily from (H8) and the Lebesgue dominated
convergence theorem. We need the structure condition (H11) in order to prove the latter.
Let us fix ǫ > 0 and show that, for all n sufficiently large, one has∫∫
Q
|fn(vn)− f(v)|p′ < ǫ. (11)
First, it follows from (H4) and Lemma 2 that |fn(vn)|p′ are equiintegrable on Q .
Indeed, take α > 0 . For all E ⊂ Q , |E| < α , we have for ε > 0 and the corresponding
Kε2
∫∫
E
|fn(vn)|p′ ≤ ε
(∫ T
0
∫
Ω
Kε2(t)Bn(vn)(t)dt+
∫∫
Q
|vn|p)+
∫∫
E
K2 ≤
≤ ε supn
(‖K1‖L1(0,T ) ‖Bn(vn)‖L∞(0,T ;L1(Ω)) + ‖vn‖pLp(0,T ;V )
)+
∫∫
E
Kε2 ,
which is independent of n and can be made as small as desired by a choice of ε and α
small enough. Fix α > 0 such that
∫∫
E
|fn(vn)− f(v)|p′ < ǫ/3 whenever |E| < α .
Further, by the Chebyshev inequality, there exists M > 0 such that for all n ∈ IN one
can choose an open set Qα,n ⊂ Q with |Qα,n| < α so that |vn| ≤ M , |v| ≤ M on
Q \Qα,n . Fix M ; for all (t, x) ∈ Q define the set Kt,xM ⊂ IRN as the image of [−M,M ]N
by b(t, x, ·) . Let TM (t, x, ·) be the projection of IRN on this set, i.e.,
TM(t, x, ·) : z ∈ IRN 7→ TM(t, x, z) = z ∈ Kt,x
M ,
where dist(z, z) = minζ∈Kt,xM
dist(z, ζ).(12)
112 Continuous Dependence for Elliptic-Parabolic Problems
The projection is well defined since, clearly, Kt,xM is compact and convex. Note that TM :
Q× IRN 7→ IRN is Caratheodory. Indeed, it is a contraction in IRN for all (t, x) ∈ Q ; for
all z ∈ IRN it is measurable in (t, x) , because b is measurable in (t, x) and TM depends
continuously on b with respect to the norm ‖ · ‖C([−M,M ]N) .
Now for all n ∈ N , for a.a. (t, x) ∈ Q , TM bn = TM(t, x, bn(t, x, ·)) is well defined.
By (H11) , one has for a.a. (t, x) ∈ Q
|fn(vn)− f(v)|p′ ≤ const( ∣∣∣fn(vn)− f(vn)
∣∣∣p′
+
+∣∣∣(f b)(vn)− (f TM b)(vn)
∣∣∣p′
+∣∣∣(f TM b)(vn)− (f TM bn)(vn)
∣∣∣p′
+
+∣∣∣(f TM )(bn(vn))− (f TM )(b(v))
∣∣∣p′
+∣∣∣(f TM b)(v)− (f b)(v)
∣∣∣p′ )
.
(13)
By Remark 3, we can apply the Lusin and Egorov theorems to fn, f and f TM bn, f TM b . Indeed, by (9) fn → f and f TM bn → f TM b in C([−M,M ]; IRN ) for a.a.
(t, x) ∈ Q . Besides, one also have bn(vn) → b(v) a.e. on Q . It follows that there exists
an open set Qα ⊂ Q with |Qα| such that one has
fn → f and f TM bn → f TM b in C((Q \Qα)× [−M,M ]N ; IRN );
bn(vn) → b(v) in C(Q \Qα; IRN );
f TM is uniformly continuous on (Q \Qα)× [−M,M ]N .
(14)
In addition, for a.a. (t, x) ∈ Q , TM b ≡ b on [−M,M ]N .
For each n , we can integrate in (11) separately over Qα ∪Qα,n and Q \ (Qα ∪Qα,n) .
The second integral vanishes as n→ ∞ , due to (13) and (14), and the first one is estimated
by 2ǫ/3 . Hence (11) holds for n small enough.
It remains to justify the passage to the limit in the elliptic term. This can be made through
a usual Minty-Browder argument summarized in Lemma 7 in Appendix. Let introduce the
operators An : η ∈ Lp(0, T ;V ) 7→ Lp′(0, T ;V ′) by defining the duality product of Anη with
ϕ ∈ Lp(0, T ;V ) :
∫ T
0
< Anη, ϕ >=
∫∫
Q
an(vn, Dη) : Dϕ.
By (H7) and Lemma 2, this last integral makes sense. Likewise, the operator A : η ∈Lp(0, T ;V ) 7→ Aη ∈ Lp′(0, T ;V ′) is defined by assigning
∫ T
0
< Aη, ϕ >=
∫∫
Q
a(v,Dη) :
Dϕ. . Since vn is a weak solution of (Pn) , it follows from the analysis above that Anvn∗ χ
in Lp′(0, T ;V ′) , where
∫ T
0
< χ, ϕ >=
∫ T
0
< −b(v)t, ϕ > +
∫∫
Q
f(v) · ϕ+
∫∫
Σ
g(v) · ϕ
for ϕ ∈ Lp(0, T ;V ) .
2.I.3. Comments and further results 113
Let verify the other assumptions of Lemma 7. We have vn v in Lp(0, T ;V ) ; all
An are monotone; besides, A is hemicontinuous by (H7) and the Lebesgue dominated
convergence theorem. Further, under the assumption (H12) , Anη → Aη in Lp′(0, T ;V ′)
for all fixed η ∈ Lp(0, T ;V ) ; the arguments are the same as used for the proof of convergence
of fn(vn) above.
Furthermore, let prove that one also has
lim inf
∫ T
0
< Anvn, vn >≤∫ T
0
< χ, v > . (15)
First, under the structure condition (H13) we can show that
∫∫Φn,t(vn) →
∫∫Φt(v) as
n→ ∞ (the last integral makes sense, by Lemma 1). Indeed, by (H13) Φt(v) = φ(b(v))+
v · ϕ(b(v)) and ϕ(b(v)) ∈ Lp′(Q) . Since vn v in Lp(Q) , it suffices to show that∫∫
Q
|Φn,t(vn)− φ(b(v))− vn · ϕ(b(v))| vanishes as n→ ∞ . This can be done by the same
arguments as used for the proof of convergence of fn(vn) . With TM(t, x, ·) defined by (12),
we have to replace the key estimate (13) by the lengthy, but trivial estimate
∣∣∣Φn,t(vn)− φ(b(v))− vn · ϕ(b(v))∣∣∣ ≤
∣∣∣Φn,t(vn)− Φt(vn)∣∣∣ +
∣∣∣φ(b(vn))− φ(b(v))∣∣∣ +
+∣∣∣vn∣∣∣∣∣∣ϕ(b(vn))− ϕ(b(v))
∣∣∣ ≤∣∣∣Φn,t(vn)− Φt(vn)
∣∣∣ +∣∣∣(φ b)(vn)− (φ TM b)(vn)
∣∣∣ +
+∣∣∣(φ TM b)(vn)− (φ TM bn)(vn)
∣∣∣ +∣∣∣(φ TM)(bn(vn))− (φ TM)(b(v))
∣∣∣ +
+∣∣∣(φ TM b)(v)− (φ b)(v)
∣∣∣ + ǫ∣∣∣vn∣∣∣p
+ C(ǫ)
∣∣∣(ϕ b)(vn)− (ϕ TM b)(vn)∣∣∣p′
+
+∣∣∣(ϕ TM b)(vn)− (ϕ TM bn)(vn)
∣∣∣p′
+∣∣∣(ϕ TM)(bn(vn))− (ϕ TM)(b(v))
∣∣∣p′
+
+∣∣∣(ϕ TM b)(v)− (ϕ b)(v)
∣∣∣p′,
where we have used the inequality |a · b| ≤ ǫ|a|p + C(ǫ)|b|p′ for a, b ∈ IRN . The required
convergence will follow from the convergences of bn,Φn,t to b,Φt , respectively, given by (9),
and the a.e. convergence of bn(vn) to b(v) . Secondly, without loss of generality we can
apply Lemma 1 with t = T to v and all functions vn , n ∈ IN . Using the convergence of
Ψn(0, u0,n) in (9), we have
lim supn→∞
∫ T
0
< −bn(vn)t, vn >= lim supn→∞
(−∫
Ω
Ψn(bn(vn))(T ) +
∫
Ω
Ψn(0, u0,n))−
−∫∫
Q
Φn,t(vn) ≤ −∫
Ω
Ψ(b(v))(T ) +
∫
Ω
Ψ(u0)−∫∫
Q
Φt(v) =
∫ T
0
< −b(v)t, v >;
the inequality here is due to the Fatou Lemma. Together with the strong convergence of
gn(vn) and fn(vn) , this yields (15). By Lemma 7, χ = Av in Lp′(0, T ;V ′) , which implies
(ii2) . ⋄
114 Continuous Dependence for Elliptic-Parabolic Problems
3 Comments and further results
3.1. On the inhomogeneous Dirichlet boundary conditions
One can deduce from Corollary 1 some existence results for non-zero Dirichlet boundary con-
ditions h ∈ Lp(0, T ;W 1,p(Ω; IRN)) . Indeed, Definition 1 permits to perform the translation
Th : v 7→ v − h ; writing down the restrictions induced by (H2)−(H13) and Th , we obtain
hypotheses that guarantee the existence of weak solutions. Within certain classes of h , the
hypotheses of Section 1 remain invariant. Let give two examples. For simplicity, assume that
K1, Kε1 ∈ L∞(0, T ) for the first case.
Corollary 2 Let (H1), (H4)−(H9), (H11)−(H13) hold, with K1 = const in (H4), (H7) ,
(H13) and Kε1 = constε in (H6) for each ε > 0 . Assume that for all z ∈ IRN , all
t ∈ [0, T ] and a.a. x ∈ Ω
|b(t, x, z)| ≤ K0(t, x)(1 + |z|p/κ
), where κ ∈ [1, p), κ′ = κ/(κ− 1)
and K0 ∈ Lν(Q) with K0(0, ·) ∈ Lν(Ω) for some ν ∈ [κ′,+∞];(H2’)
assume that for all ε > 0 there exists Kε2 ∈ L1(Q) such that for all z ∈ IRN and a.a.
(t, x) ∈ Q
|Φt(t, x, z)| ≤ ε|z|p +Kε2(t, x). (H3’)
Let 1/σ + 1/ν + 1/κ = 1 . Then for all Dirichlet data h ∈ Lp(0, T ;W 1,p(IRN )) such that
h ∈ W 1,σ(0, T ;Lσ(Ω; IRN)),
h(·, x) is absolutely continuous on [0, T ] for a.a. x ∈ Ω
with h(0, ·) ∈ Lp(Ω; IRN) and h(0, ·) · u0(·) ∈ L1(Ω),
(H10’)
there exists a weak solution to (P ) .
Proof of Corollary 2: It is equivalent to consider instead of (P ) the problem (P )
with the zero Dirichlet data and b(t, x, z) = b(t, x, z + h(t, x)) , a(t, x, z, ξ) = a(t, x, z +
h(t, x, ), ξ +Dh(t, x)) , f(t, x, z) = f(t, x, z + h(t, x)) and g(t, x, z) = g(t, x, z + h(t, x)) .
Note that (H1) holds with Φ(t, x, z) = Φ(t, x, z + h(t, x)) − Φ(t, x, h(t, x)) , so that
Ψ(t, x, z∗) = Ψ(t, x, z∗)− h(t, x) · z∗ + Φ(t, x, h(t, x)) and
B(t, x, z) = B(t, x, z + h(t, x))− h(t, x) · b(t, x, z + h(t, x)) + Φ(t, x, h(t, x)). (16)
Clearly, the initial condition is unchanged: u0 = u0 on Ω .
Since h(0, ·) ∈ Lσ(Q; IRN) by the Fatou lemma, and because
|Φ(0, x, h(0, x))| ≤ K0(0, x) |h(0, x)| (1 + |h(0, x)|p/κ) ≤≤ const
((K0(0, x))
ν + |h(0, x)|σ + 1 + |h(0, x)|p),
2.I.3. Comments and further results 115
it follows from (H2′) and (H10′) that Ψ(u0) ∈ L1(Ω) . Hence (H9) holds. The hy-
pothesis (H2) is satisfied because K0 |h|p/κ ∈ L1(Q) . Further, (H5), (H11), (H12) are
obvious. Since h,Dh ∈ Lp(Q) , the invariance of (H8) follows by the Holder inequality;
moreover, checking (H4), (H6), (H7) amounts to showing that
B(t, x, z + h(t, x)) ≤ B(t, x, z) + C|z|p +K2(t, x) (17)
with some function K2(t, x) ∈ L1(Q) . Since Φ is non-negative, one has by (H2′) and
(16)
B(t, x, z + h(t, x)) ≤ B(t, x, z) + h(t, x) · b(t, x, z + h(t, x)) ≤≤ B(t, x, z) + |h(t, x)| |K0(t, x)| (1 + |z + h(t, x)|p/κ),
(18)
whence (17) follows.
It remains to show that (H3) and (H13) hold for (P ) as well. One has
Φt(t, x, z) = ∂/∂t h(t, x) ·(b(t, x, z + h(t, x))− b(t, x, h(t, x)
)+
+Φt(t, x, z + h(t, x))− Φt(t, x, h(t, x)).(19)
Hence Φt(t, x, z) = (φ)(t, x, b(t, x, z)) + z · (ϕ)(t, x, b(t, x, z)) with (ϕ) ≡ ϕ . As in the
estimate (18), using in addition the inequality c|a · b| ≤ C(ǫ)cν +C(ǫ)|a|σ+ ǫ|b|κ valid for all
c ∈ IR+, a, b ∈ IRN , one sees that (H3′) and the inequality in (H13) are invariant under
the translation by h , whenever (H2′), (H10′) hold.
Therefore (P ) is in the scope of Corollary 1. ⋄
Remark 4 Clearly, under the assumptions (H1), (H2′), (H3′), (H4)−(H9), (H11)−(H13)
on (P ) , for the class of Dirichlet data h verifying (H10′) there is also a continuous depen-
dence on h of weak solutions of (P ) in the sense of Theorem 1.
In case without dependence of coefficients on (t, x) , the assumptions on h of Corollary 2
meet exactly the assumptions in the remark in [AL83] that follows Lemma 1.5. Nevertheless,
if we could modify the condition (H3) by excluding s 6= t (cf. Remark 6 in the Appendix),
there would be no need to strengthen (H3) to (H3′) . The problem here is that, in what
concerns s 6= t , (H3) is utterly non-invariant under translations Th with natural assump-
tions on h . For instance, with the hypothesis (H3) we cannot attain the assumptions on h
made in the statement of Lemma 1.5 in [AL83] unless requiring (H2′′) below, which imposes
a growth restriction in z †. We state the corresponding result in case where b is independent
of t , so that (H3) does not seem pecular.
†This hypothesis is verified up to exponential growthes of b(x, z) in z
116 Continuous Dependence for Elliptic-Parabolic Problems
Corollary 3 Let b be independent of t , and (H1), (H3)−(H9), (H11), (H12) hold. As-
sume
for all r ∈ IR+ there exist Λ(r) <∞ and K0(·, r) ∈ L1(Ω)
such that for all λ, z ∈ IRN , and a.a. x ∈ Ω one has
|b(x, z + λ)| ≤ Λ(|λ|)|b(x, z)|+ K0(x, |λ|) and |b(x, 0)| ≤ K0(x, 0).
(H2”)
Then there exists a weak solution of (P ) for all Dirichlet data h ∈ Lp(0, T ;W 1,p(IRN)) such
that
h ∈ L∞(Q; IRN), ∂/∂t h ∈ L1(0, T ;L∞(Ω; IRN ))
and h(·, x) is absolutely continuous on [0, T ] for a.a. x ∈ Ω.(H10”)
Proof of Corollary 3: We proceed as in the proof above. First note that it follows
from (H2′′) that |b(x, z)| ≤ K0(x, |z|) = Λ(|z|)K0(x, 0)+K0(x, |z|) . Let M = ‖h‖L∞(Q) .
Clearly, |b(t, x, z)| ≤ K0(x, |z| +M) , so that (H2), (H9) hold for (P ) . Besides, one has
to show that
|Φt(t, x, z)| ≤ εK1(t)B(s, x, z) +Kε2(t, x) (20)
for all z ∈ IRN , a.a. t, s ∈ (0, T ) and a.a. x ∈ Ω , where K1 ∈ L1(0, T ) and Kε2(t, x) ∈
L1(Q) , and that
B(x, z + h(t, x)) ≤ B(t, x, z) +K2(t, x) (21)
for all z ∈ IRN and a.a. (t, x) ∈ Q , where K2 ∈ L∞(0, T ;L1(Ω)) .
First, by (H2′′) and (7) with δ = ε/Λ(2M) one has
|b(x, z + h(t, x))| ≤ Λ(2M)|b(x, z + h(s, x))|+ K0(x, 2M) ≤ Λ(2M)|b(s, x, z)|++K0(x, 2M) ≤ εB(s, x, z) +K0(x, ε/Λ(2M) +M) + K0(x, 2M).
Hence it follows from (19), with Φt ≡ 0 , that (20) holds with K1(t) = ‖∂/∂t h(t, ·)‖L∞(Ω)
and Kε2(t, x) = |∂/∂t h(t, x)|
(K0(x,M) +K0(x, ε/Λ(2M) +M) + K0(x, 2M)
).
Further, (21) holds, because by (16) and (7) with δ = 1 one has
B(x, z + h(t, x)| ≤ B(t, x, z) + |h(t, x)| |b(t, x, z)| ≤ (1 +M)B(t, x, z) + K0(x,M + 1). ⋄
3.2. On the structure conditions
Let us turn now to the relevancy of the hypotheses (H11)−(H13) . First note one direct
generalization of Theorem 1 and Corollaries 1,2,3.
2.I.3. Comments and further results 117
Remark 5 An additional term of the form f(·, v) can be considered in the right-hand side
of (P ) , provided f : Q× IRN 7→ IRN is Caratheodory and satisfy
(f(t, x, z)− f(t, x, z)) · (z − z) ≤ 0,
|f(t, x, z)|p′ ≤ C |z|p +K1(t)B(t, x, z) +K2(t, x)(H14)
for all z, z ∈ IRN , for a.a. (t, x) ∈ Q , with some K2 ∈ L1(Q) and C = const ≥ 0 . For
(P ) modified in such a way, (H) and (9) modified correspondingly, and (H14) admitted,
Theorem 1 and Corollary 1 still hold. Indeed, in this case we can include the terms −fn(t, x, ·)into the operators An , and An remain monotone. Therefore we do not need a structure
condition of the kind (H11) on f .
Besides this remark, it is easy to see that in case the dependence on z of f and a is
weak, (H11) and (H12) can be superfluous while proving the existence of a weak solution.
Indeed, for p = 2 consider the model problem
b(v)t = ∆v + divF (v) on Q = (0, T )× Ω (MP)
with F : IRN 7→ IRN which is globally Lipschitz continuous, and with the homogeneous
Dirichlet boundary condition (so that V = W 1,p0 (Ω; IRN ) ). The operator A : η ∈ L2(0, T ;V )
7→ L2(0, T ;V ′) defined by∫ T
0
< Aη, ϕ >=
∫∫
Q
Dη : Dϕ+ F (η) ·Dϕ
for all ϕ ∈ L2(0, T ;V ) is monotone provided the diameter of Ω is small, due to the Poincare
inequality. Therefore we can use Lemma 7 with An ≡ A while passing to the limit in a
sequence of approximating problems (MPn) with bn bilipschitz, and with solutions vn such
that vn v in L2(0, T ;V ) . This yields that Avn Av in L2(0, T ;V ′) , but also that
vn → v strongly in L2(0, T ;V ) ; so one can pass to the limit in each term in (MP ) and
infer the existence.
For scalar problems of type (MP ) , an existence result for arbitrary locally Lipschitz
continuous F has been shown in [BW99]. The methods used in this paper are those of the
nonlinear semigroup theory; unfortunately, this gives no information on the type of convergence
of approximating solutions to the weak solution of (MP ) .
Practically, (H11) , (H12) often hold in problems motivated by physics. For instance, in
the Richards equation (cf. [Bear]) v and u represent the liquid pressure and the saturation of
the medium, respectively; the nonlinearity in the right-hand side of the corresponding equation
is the permeability of the medium which is a function of u . On the other hand, the condition
(H13) is proper to the function b(·) and seems to impose restrictions on its dependence on
t . While the author was not able to prove that the representation
Φt(t, x, z) = φ(t, x, b(t, x, z)) + z · ϕ(t, x, b(t, x, z))for all z ∈ IRN and a.a. (t, x) ∈ Q
(22)
118 Continuous Dependence for Elliptic-Parabolic Problems
with φ, ϕ Caratheodory generically holds, the following result fixes the situation in the scalar
case.
Lemma 3 Let N = 1 . Assume that bt is Caratheodory. Then (22) holds, and one can take
ϕ such that for all z ∈ IRN and a.a. (t, x) ∈ Q one has ϕ(t, x, b(t, x, z)) = bt(t, x, z) .
Proof: The dependence on x is immaterial here, and we neglect it in the notation.
Let first prove that, for a.a. t0 ∈ (0, T ) , if Φ(t0, ·) is affine in an interval [z, z] (i.e.,
b(t0, ·) degenerates) it follows that its derivative with respect to t is also affine on [z, z] .
Indeed, assume that this property fails on some subset B of (0, T ) with |B| > 0 . For each
t0 ∈ B there exist two rational points z, z ∈ [z, z] such that the function
∆(t0, z, z; ·) = (z − z)(Φ(t0, ·)− Φ(t0, z))− (· − z)(Φ(t0, z)− Φ(t0, z))
is identically zero on [z, z] , while the continuous function
∇(t0, z, z; ·) = (z − z)(Φt(t0, ·)− Φt(t0, z))− (· − z)(Φt(t0, z)− Φt(t0, z))
is non-zero at some rational point z0 ∈ (z, z) . Still ∇(t0, z, z; z0) = ∆t(t0, z, z; z0) . Hence
B ⊂ ⋃Bz,z,z0 , where Bz,z,z0 = t0 ∈ (0, T ) | ∆(t0, z, z; z0) = 0, but ∆t(t0, z, z; z0) 6= 0 ,
and the union is taken over all z, z, z0 ∈ IQ . Since ∆(·, z, z; z0) is absolutely continuous on
[0, T ] , |Bz,z,z0| = 0 for all z, z, z0 ∈ IQ , which is a contradiction.
Note that for t0 /∈ B , one has bt(t0, ·) = (Φt(t0, z) − Φt(t0, z))/(z − z) = const on
[z, z] . Hence we have shown that for a.a. (t, x) ∈ Q , all z, z ∈ IRN one has
b(t, z) = b(t, z) ⇒
bt(t, z) = bt(t, z) and
Φt(t, z)− Φt(t, z) = (z − z)bt(t, z)
for all z between z and z.
(23)
This is equivalent to the statement of the lemma. Indeed, (23) means that bt(t, ·) is in
fact a function of b(t, ·) : bt(t, z) = ϕ(t, b(t, z)) . The function ϕ(t, ·) is continuous. Indeed,
let βn, β ∈ R(b(t, ·)) and βn → β as n → ∞ . Due to the monotony of b(t, ·) , thereexist zn ∈ b(t, ·)−1(βn) , z ∈ b(t, ·)−1(β) such that zn → z . Hence ϕ(t, βn) = bt(t, zn) →bt(t, z) = ϕ(t, β) as n→ ∞ . Furthermore, by (23), Φt(t, ·)− z ϕ(t, b(t, ·)) stays constant
as b(t, ·) degenerates. By the same argument, it can be written under the form φ(t, b(t, ·)) ,with φ(t, ·) continuous. ⋄
3.3. On pure Neumann boundary conditions for one or more components
The restriction |Γ0| > 0 , imposed in Sections 1,2, can be relaxed in two ways. The first
one consists in imposing hypotheses that provide a priori estimates on ‖Dv‖Lp(Q) and
‖B(v)‖L∞(0,T ;L1(Q)) , and another hypothesis that permits to control |v| by |B(v)| . This
requires some modification of our arguments, and we do not pursue this line here. On the
2.I.4. Appendix 119
other hand, some cases can be included by requiring directly the coercivity of the elliptic part
in the space V for all t ∈ (0, T ) ; more precisely, it is sufficient (in case h ≡ 0 ) that, instead
of (H6) and the second part of (H8) , the following assumption be fulfilled:∫
Ω
a(t, ·, w(·), Dw(·)) : Dw(·)− f(t, ·, w(·)) · w(·) + Φt(t, ·, w(·))
−
−N∑
i=1
∫
Γ1,i
gi(t, ·, w(·))wi(·) ≥ c‖w‖V −K1(t)B(t, ·, w(·))−K1(t)
for all w ∈ V and a.a. t ∈ (0, T ) , where K1 ∈ L1(0, T ) and c > 0 . In this case, Lemma 2
is immediate, and other arguments run without any modification.
3.4. On the convergence of approximate methods
Note that Theorem 1 can be used to prove the convergence while solving (P ) by various
approximate methods. One can almost directly apply the theorem to Galerkin approximations.
Indeed, the equation for Galerkin approximations is (P ) itself with an error term that tends
to zero weakly in Lp′(0, T ;V ′) , which still permits to apply the Minty-Browder argument
for convergence. For finite volumes methods, an example is given in [BAGuW] and Chapter
2.II, where the convergence for a class of schemes is proved for a simple model elliptic-
parabolic system. The steps of the proof of Theorem 1 are the same, provided the system that
determines the discret solution is rewritten under the form of equation in D′ similar to (P ) .
Approximations by time implicit discretization (at least, with uniform step) can be treated in
a similar way; in this case writing the system that determines the solution under the form of
equation in D′ is trivial.
Appendix
Here we present the appropriate versions of the three essential arguments involved in the proof
of Theorem 1.
A1. The chain rule argument
Let use the shortened notation w(t) for w(t, x) , b(w)(t) for b(t, x, w(t, x)) , Φt(w)(t) for
Φt(t, x, w(t, x)) , Φt(τ, w(t)) for Φt(τ, x, w(t, x)) and so on. Let start with two auxiliary
results.
Lemma 4 For k > 0 , let Tk : z ∈ IRN 7→ min|z|, k z/|z| ∈ IRN . Assume that b satisfy
(H1) and Φ is absolutely continuous on [0, T ] for all z ∈ IRN and a.a. x ∈ Ω . Then one
has (omitting the dependence in x )
B(t, Tk(z)) ≤ B(s, z) + (b(t, z)− b(s, z)) · Tk(z)−∫ t
s
Φt(τ, Tk(z)) dτ
for all z, z ∈ IRN , t, s ∈ [0, T ] , x ∈ Ω .
120 Continuous Dependence for Elliptic-Parabolic Problems
Proof: By the convexity of Φ in z , one has
B(t, Tk(z))− B(s, z) = B(s, Tk(z))−B(s, z) +B(t, Tk(z))− B(s, Tk(z)) ≤≤ (b(s, Tk(z))− b(s, z)) · Tk(z) + b(t, Tk(z)) · Tk(z)− b(s, z) · Tk(z)−
−Φ(t, Tk(z)) + Φ(s, Tk(z)) = (b(t, Tk(z))− b(s, z)) · Tk(z)−∫ t
s
Φt(τ, Tk(z)) dτ.
Since b is monotone and z−Tk(z) = (|z|/k−1)+Tk(z) , it follows that b(t, Tk(z)) ·Tk(z) ≤b(t, z) · Tk(z) . ⋄
Lemma 5 Let φ : Q × IRN 7→ IR be Caratheodory. Assume that for all z ∈ IRN , for a.a.
(t, x) ∈ Q one has
|φ(t, x, z)| ≤ F (t, x, |z|) with F (·, x, r) ∈ L1(0, T ) for all r > 0. (24)
For h 6= 0 , set φh(t, x, z) = 1/h
∫ t
t−h
φ(τ, x, z) dτ , where φ(·, z) is extended by 0 outside
Q . Assume that v : Q 7→ IRN is measurable and for all h one has
|φh(v)(·)| ≤ F (·) for all h, where F ∈ L1(Q). (25)
Then φh are Caratheodory and φh(v) → φ(v) in L1(Q) .
Proof: By (24), for all h > 0 , for all bounded set K in IRN , |φh(·, z)| are dominated
by a fixed function in L1(Q) for all z ∈ K . Therefore φh are Caratheodory.
By Remark 3 it follows that φ, φh are strongly measurable from Q to C(IRN) . Moreover,
φh → φ a.e. on Q in C(IRN) . Therefore for all sequence hk → 0 one can apply the Lusin
and Egorov theorems to φhk . Thus for all α > 0 there exists an open subset Qα of Q
with |Qα| < α such that φhk → φ in C(Q \Qα;C(IRN)) = C(Q \Qα × IRN) .
By (25), φhk(v) are equiintegrable on Q , so that it is sufficient to prove the lemma in
case v(·) is bounded on Q by some constant M and φhk , φ are equicontinuous in (t, x, z)
on Q × [−M,M ]N . Let approximate v a.e. on Q by a sequence of functions vm each
taking a finite number of values in [−M,M ]N . By the diagonal process, one can extract a
subsequence hkn such that φhkn (vm) → φ(vm) a.e. on Q for all m ∈ IN . Take (t, x) in
the set where this convergence takes place. For each m fixed, one has
|φhkn (v)− φ(v)| ≤ |φhkn (v)− φhkn (vm)|+ |φhkn (vm)− φ(vm)|+ |φ(vm)− φ(v)|,
which can be made as small as desired by a choice of m and n large enough. Since the
sequence hk → 0 is arbitrary, the assertion of the lemma follows by the Lebesgue theorem.
⋄
Proof of Lemma 1: Take v0 measurable such that b(v0) = u0 , and extend v(t) by v0
to t < 0 . One has b(v)t = −χ ≡ 0 for t < 0 .
2.I.4. Appendix 121
First, take t ∈ (0, T ) , h > t − T and apply Lemma 4 to s = t − h , z = v(t) ,
z = v(t− h) . Denote Tk(v(·)) by vk(·) ; note that ‖vk(t)‖V ≤ ‖v(t)‖V . After integration
in x , one obtains∫B(vk)(t) ≤
∫B(v)(t− h)+
+
∫(b(v)(t)− b(v)(t− h)) · vk(t)−
∫ t
t−h
∫Φt(τ, vk(t)) dτ.
(26)
By (ii1) of Definition 1 and (H3) we deduce
∫ T
0
(∫B(v)(t)−
∫B(v)(t− h)
)dt ≤ const |h|
∫ T
0
1h
∫ t
t−h
‖χ(τ)‖V ′ dτ ‖v(t)‖V +
+1
h
∫ t
t−h
K1(τ) dτ
∫B(v)(t) +
∫|v(t)|p +
1
h
∫ t
t−h
∫Kε
2(τ) dτdt ≤
≤ const |h|‖χ‖Lp′(0,T ;V ′) ‖v‖Lp(0,T ;V ) + ‖K1‖L1(0,T )×
×‖B(v)‖L∞(0,T ;L1(Ω)) + ‖v‖pLp(Q) + ‖Kε2‖L1(Q)
≤ const |h|.
Therefore
∫B(v) ∈ W 1,1(0, T ) . In particular, there exists l = lim
h↓0
∫B(v)(h) . We have
to show that l =
∫B(v0) . Take t = 0 , z = v0 , s = h , z = v(h) in Lemma 4. By
Definition 1 (ii1) and (H3) one has
∫B(v)(h) ≥
∫B(Tk(v0))−
∫ h
0
(u0 − b(v)(h)) · Tk(v0)−∫ h
0
∫Φt(τ, Tk(v0)) dτ ≥
≥∫B(Tk(v0)) −
∫ h
0
‖χ(τ)‖V ′ dτ ‖Tk(v0)‖V −∫ h
0
K1(τ) dτ×
×‖B(0, Tk(v0))‖L1(Ω) − ‖Tk(v0)‖pLp(Q) −∫ h
0
‖Kε2(τ)‖L1(Ω) dτ.
As h ↓ 0 and k → +∞ , one gets l ≥∫B(v0) by the Fatou lemma. The inverse inequality
can be obtained in a similar way from (26) with t = h .
To prove (8), it remains to show that
d
dt
∫
Ω
B(v) = − < χ(t), v(t) > −∫
Ω
Φt(v)(t).
By (26), it suffices to show that there exists a sequence hn ↓ 0 such that
1
hn
∫(b(v)(t)− b(v)(t∓ hn)) · v(t) =
1
hn< b(v)(t)− b(v)(t∓ hn), v(t) >→ ± < χ(t), v(t) >
in L1(Q) , and
1
hn
∫ t
t∓hn
∫Φt(τ, v(t)) dτ → ±
∫Φt(v)(t)
122 Continuous Dependence for Elliptic-Parabolic Problems
in L1(0, T ) . Since (b(v)(·) − b(v)(· ∓ hn))/hn → ±χ(·) in Lp′(0, T ;V ′) , the former
convergence is clear. Besides, the latter one can be proved by applying Lemma 5 to φ = Φt .
Indeed, (25) follows from (H3) ; (24) follows from (H3) and the continuity of B in (t, z) .
This ends the proof. ⋄
Remark 6 Formally, the formula (8) would make sense under the weaker version of (H3) :
|Φt(t, x, z)| ≤ ε(K1(t)B(t, x, z) + |z|p
)+Kε
2(t, x). (H3?)
for a.a. (t, x) ∈ Q and all z ∈ IRN . On the other hand, except in the proof above, we
actually need this weaker form whenever (H3) is used. At the present stage, it is not clear to
the author whether the analogue of Lemma 1 does hold when (H3) is replaced by (H3?) .
A2. The compactness argument
In his paper [K69a] Kruzhkov has proved that, for a bounded weak solution of a general
evolution equation, the L1 -modulus of continuity in t can be estimated via the L1 -modulus
of continuity in x and an a priori bound on the right-hand side in some Sobolev space
L1(0, T ;W−m,1
). Here we present this lemma, with somewhat less restrictive assumptions,
as a kind of compactness result.
Lemma 6 Let Ω be an open subset of IRN , E be a set of indexes, and uε ∈ L1((0, T )×Ω ,
ε ∈ E , satisfy
∂
∂tuε(t, x) =
∑
|α|≤m
(−1)αDαAεα(t, x) in D′((0, T )× Ω).
Assume that for all compact set K ⊂ Ω the following estimates hold uniformly in ε :
(i) ‖uε‖L1((0,T )×K) ≤ CK ;
(ii)∑
|α|≤m ‖Aεα‖L1((0,T )×K) ≤ CK ;
(iii) for all h > 0 small enough sup|∆x|≤h
∫ T
0
∫
K
|uε(t, x+∆x)− uε(t, x)|dxdt ≤ ωK(h) ,
with limh→0 ωK(h) = 0 .
Then the family uεε∈E is relatively compact for the L1loc((0, T )× Ω) -topology.
Proof of Lemma 6: We only need to find a function ωK on IR+ such that limτ→0
ωK(τ) = 0
and I :=
∫ T−τ
0
∫
K
|uε(t + ∆t, x) − uε(t, x)|dxdt ≤ ωK(τ) for all ∆t ∈ [0, τ ] and ε ∈ E .
Indeed, the result will follow from (i),(iii) and the Kolmogorov theorem.
2.I.4. Appendix 123
Let revise the proof from [K69a]. Fix ε ∈ E , ∆t ∈ [0, τ ] . For t ∈ [0, T − τ ] set
wt(·) := uε(t+∆t, ·)− uε(t, ·) . For all ϕ ∈ D(K) we have∫
K
wt(x)ϕ(x)dx =
∫ t+∆t
t
∫
K
∑|α|≤m
Aεα(θ, x)D
αϕ(x)dxdθ. (27)
For each t ∈ [0, T−τ ] take for the test function ϕ(·) a regularisation of signwt(·) . More
precisely, let δ := dist(K, ∂Ω) and h ∈ (0, δ/2) . Set K−h = x ∈ K : dist(x, ∂K) ≥ hand denote by χK−h
(·) its characteristic function. Choose ρ ∈ C∞0 ([−1, 1]N) , ρ ≥ 0 such
that
∫ρ(σ)dσ = 1 , and take in (27) ϕt(x) = h−N
∫ρ(x− y
h
)signwt(y)χK−h
(y)dy . It
is clear that for |α| ≤ m , ‖Dαϕt‖L1([0,T ]×K) ≤ const · h−m uniformly in t and ε .
Note that I = I1+I2 , where I1 =
∫ T−τ
0
∫
K
wt(x)ϕt(x)dxdt and I2 =
∫ T−τ
0
∫
K
(|wt(x)|−
wt(x)ϕt(x))dxdt . Using (27) and (ii), we get |I1| ≤ const ·τh−m
∑|α|≤m ‖Aε
α‖L1([0,T ]×K) ≤
CKτ/h−m . Besides, by (i),(iii) and the Kolmogorov theorem the family
∫ T
0
|uε(t, ·)|dtε∈E
is relatively compact in L1(K) . Therefore these functions are equiintegrable on K , so that∫ T
0
∫
K\K−2h
|uε(t, x)|dxdt ≤ ωK(h) with limh→0 ωK(h) = 0 uniformly in ε . Hence
|I2| ≤ 2
∫ T−τ
0
∫
K\K−2h
|wt(x)|dxdt+
+
∫ T−τ
0
∫
K−2h
∣∣∣|wt(x)| − wt(x)
∫h−Nρ
(x− y
h
)signwt(y)
∣∣∣dydxdt ≤ 4ωK(h) +
+
∫ T−τ
0
∫
K−2h
∫h−Nρ
(x− y
h
) ∣∣∣|wt(x)| − wt(x) signwt(y)∣∣∣dydxdt.
Since for all a, b ∈ IR we have ||a| − a sign b| ≤ 2|a− b| , it follows that
|I2| ≤ 4ωK(h) + 2
∫ T−τ
0
∫
K−2h
∫h−Nρ
(x− y
h
)|wt(x)− wt(y)|dydxdt ≤
≤ 4ωK(h) + 4
∫ρ(σ)
∫ T
0
∫
K−2h
|uε(t, x)− uε(t, x− hσ)|dxdtdσ ≤ 4ωK(h) + 4ωK(h).
The function ωK(τ) = min0<h≤δ/2
CK
τhm + ωK(h) + ωK(h)
majorates I and tends to 0 as
τ → 0 , which ends the proof. ⋄
A3. The Minty-Browder argument
Lemma 7 Let E be a Banach space, E ′ its dual and (·, ·) denote the duality product of
elements of E ′ and E . Take a sequence vn in E such that vn v . Take a sequence
of monotone operators An : E 7→ E ′ such that An converge pointwise to some operator
A : E 7→ E ′ and Anvn∗ χ in E ′ .
Then χ = Av whenever A is hemicontinuous (i.e., continuous in the weak-∗ topology
of E ′ along each direction) and lim infn→∞(Anvn, vn) ≤ (χ, v) .
124 Continuous Dependence for Elliptic-Parabolic Problems
Proof of Lemma 7: The proof is standard (e.g., see [JLL]); we give it here for the sake
of completeness. Under the assumptions of Lemma 7, for all η ∈ E one has
(χ, v − η) ≥ lim infn→∞
(Anvn, vn − η) ≥ lim infn→∞
(Anη, vn − η) = (Aη, v − η).
Taking η = v + λζ with λ ∈ IR , ζ ∈ E and letting λ increase to zero, one gets (χ, ζ) ≥(Av, ζ) . As λ decreases to zero, the inverse inequality follows, so that (χ, ζ) = (Av, ζ) for
all ζ ∈ E . ⋄
CHAPTER 2.II
Convergence of Finite Volumes Approximations
for a Nonlinear Elliptic-Parabolic Problem:
a Variational Approach†
Introduction
Let Ω be an open bounded polygonal domain in Rd , d ≥ 1 and T > 0 . We consider the
initial boundary value-problem for a system of nonlinear elliptic-parabolic equations:
b(v)t = div ap(Dv) on Q = (0, T )× Ω,
v = 0 on Σ = (0, T )× ∂Ω,
b(v)(0, ·) = u0 on Ω,
(1)
where 1 < p <∞ and div ap(Dv) = div (|Dv|p−2Dv) is the N -dimensional p -Laplacian,
N ≥ 1 , i.e.,
ap : ξ = (ξ1, . . . , ξN) ∈ (Rd)N 7→ |ξ|p−2ξ =(∑i,j |ξji |2
)p/2−1
(ξ1, . . . , ξN) ∈ (Rd)N .(2)
We assume that
b : RN → RN is continuous, monotone with b(0) = 0, i.e.,
there exists a convex differentiable function Φ : RN → R
with Φ(0) = 0 such that b = ∇Φ,
(3)
and
u0 ∈ L1(Ω)N with Ψ(u0) ∈ L1(Ω), (4)
where Ψ is the Legendre transform of Φ given by
Ψ : z ∈ RN 7→ sup
σ∈RN
∫ 1
0
(z − b(sσ)) · σ ds.
†The results of this chapter are being prepared upon publication [BAGuW]
126 Variational Approach for a Finite Volume Method
Systems of elliptic-parabolic equations of type (1) arise as a model of flow of (several) fluids
through porous media (cf. e.g. [Bear, DiDT94]). They have already been studied extensively
in the literature in the last decade from a theoretical point of view (cf. e.g. [ALpr, AL83,
Kac90, DiDT94, BW96, Ot96, Bou97, BW99, CaW99, BAB]). Existence of weak solutions of
general systems of elliptic-parabolic equations has been proved in [ALpr, AL83], using Galerkin
approximations and time-discretization. Similar results have been obtained later by other
authors using different methods (e.g., using a semigroup approach as in [BW96, Bou97] in the
case N = 1 ).
In particular, it is known that in the case of the system (1), for any u0 satisfying (4),
there exists a weak solution of (1), where the weak solution is defined as follows.
Definition 1 (Weak solution) A function v ∈ E = Lp(0, T ;W 1,p0 (Ω))N is a weak solution
of the problem (1), if b(v) ∈ L∞(0, T ;L1(Ω))N , the function b(v)t ∈ E ′ = Lp′(0, T ;W−1,p′(Ω))N
(where p′ denotes the conjugate exponent of p ) satisfies
< b(v)t, φ >E′,E +
∫ ∫
Q
ap(Dv) ·Dφ = 0 (5)
for all φ ∈ E , where < ·, · >E′,E denotes the duality pairing between E ′ and E , and
− < b(v)t, ξ >E′,E=
∫ ∫
Q
b(v)ξt +
∫
Ω
u0ξ(0) (6)
for all ξ ∈ E with ξt ∈ L∞(Q)N and ξ(T ) = 0 .
Moreover, if v is a weak solution of (1), then, by the “chain rule” lemma of [AL83],
B(v) ∈ L∞(0, T ;L1(Ω))N , where
B : z ∈ RN 7→ b(z) · z − Φ(z) ≡
∫ 1
0
(b(z)− b(sz)) · zds ≡ Ψ(b(z)) ∈ R. (7)
From the results of [Ot96, CaW99] it also follows that, in the scalar case N = 1 , there is
uniqueness of a weak solution of (1). To our knowledge, the question of uniqueness is open
in the case N ≥ 2 .
The variational approach of [AL83] for elliptic-parabolic problems has been revisited in
Chapter 2.I. Beyond an extension of different existence results, in Chapter 2.I a concise variant
of the techniques of [AL83], applied to prove continuity of weak solutions of general elliptic-
parabolic problems with respect to the data and coefficients, has been presented. In this
chapter we are interested in proving, with the same techniques, the convergence of approxi-
mations by finite volumes numerical schemes for the model nonlinear elliptic-parabolic problem
(1).
Finite volumes methods are well suited for numerical simulation of processes where exten-
sive quantities are conserved, and it is a very popular method among engineers in hydrology
where systems of equations of this type arise. Therefore justification of convergence of this
2.II.0. Introduction 127
numerical approximation process is of particular interest. In [EGH98] the finite volume meth-
ods has been studied and convergence of this approximation procedure has been proved for
problem (1) in the particular case p = 2 , N = 1 . The same method has also been studied
for this equation (i.e. p = 2, N = 1 ) in the presence of an additional convection term (cf.
[EGH99]), and for a nonlinear diffusion problem in [EGHNS98].
In this chapter we deal with convergence for time implicit finite volume approximations of
(1). Let us emphasize that our main object is not only to prove the convergence of the finite
volume method for (1), but to develop the variational approach for this proof. The main idea of
this adaptation is to rewrite the discrete finite volume scheme under an equivalent continuous
form and to apply known stability techniques for the continuous equation (cf. [AL83, BAB])
in order to get convergence of the discrete approximation scheme.
In Section 1, we introduce the finite volume scheme for (1). We specify a class of admissible
partitions T h of Ω and (T h, kh) of Q (cf. Definitions 2 and 3) and define the approximate
discrete problems (Ph) ( h > 0 being the discretization parameter). To this end we have to
introduce a finite volume approximate Dh of the gradient operator D . A class of admissible
discrete gradient approximations is defined (see (14),(16) and Definition 4 below) and an
example of an admissible gradient approximation is given (cf. Remark 2).
In Section 1 we also state the main result of the chapter: for any admissible family of
grids T h, kh and gradient approximations Dh , there exists a sequence h→ 0 such that
solutions vh of the discrete problems (Ph) (cf. (24)-(26)) converge weakly in L1(Q)N to
a weak solution of (1).
In Section 2, existence of solutions vh of the discrete equations is established. Note that,
at least for special choices of the gradient approximation, it is also possible to prove uniqueness
of a discrete solution. We derive a discrete analog of Lp(0, T ;W 1,p0 (Ω))N a priori estimate
for vh and obtain a bound of B(vh) in L∞(0, T ;L1(Ω))N .
Section 3 is devoted to the problem of rewriting the discrete scheme under the form of an
equivalent equation
uht = div ap(Dvh) (8)
in D′(Q) for appropriate approximations uh ∈ L1(Q)N of b(vh) and Dv ∈ Lp(Q)N of
Dhvh .
In Section 4 we give the proof of the main theorem. The proof is essentially based on
three arguments: the a priori estimates obtained by using the chain rule argument of [AL83], a
lemma of Kruzhkov (cf. [K69a]) to get strong compactness in the parabolic part, and a Minty-
Browder argument (cf. e.g. [JLL]) to get convergence in the elliptic part of the continuous
form of the approximate equations.
128 Variational Approach for a Finite Volume Method
Sections 5 and 6 are devoted to the proof of several auxiliary results and technical details
used in the proof of the main theorem. In particular, in Section 5, in addition to the contin-
uous approximation Dvh of the discrete gradients Dhvh defined in Section 3, we construct
auxiliary continuous approximations vh of the discrete solutions vh themselves. In this step,
it is convenient to impose the assumption (10) of proportionality of the mesh. Moreover, this
assumption is essentially used in the proof of the discrete Poincare inequality (cf. Lemma 9 in
the Appendix) when p > 2 .
Section 7 is devoted to the proof of consistency of the finite volume approximation of
the elliptic operator in (1) in the sense of Definition 5 (cf. Section 7). Here we develop an
additional series of arguments. The restriction (iv) of Definition 4 (cf. Section 1) is used at
this stage.
In the Appendix we have collected several auxiliary results form the theory of Sobolev
spaces and their discrete analogues.
Throughout the chapter we keep the notations introduced in this section. Moreover, as an
abuse of notation, in the following sections, the spaces Lp(Q)N , Lp(Ω)N etc. are written
as Lp(Q) , Lp(Ω) etc.
1 The numerical scheme
In order to construct approximate solutions to Problem (1), we use in this section the implicit
discretization in time and a finite volume scheme in space.
First we introduce a notion of admissible mesh of Ω (see also [EyGaHe] and [EGH99]).
Definition 2 (Admissible mesh) Let Ω be an open bounded polygonal subset of Rd . An
admissible finite volume mesh T of Ω is given by : a family of open polygonal convex
subsets with positive measure of Ω called ”control volumes” (for the sake of simplicity, we
shall denote by T the family of control volumes); a family E of subsets of Ω contained in
hyperplans of Rd , with positive (d−1) -measure (these are the edges of the control volumes);
a family of points of Ω , where these families satisfy the following properties:
(i) The closure of the union of all the control volumes is Ω ;
(ii) For any (K,L) ∈ (T )2 with K 6= L , either the length of K ∩ L is 0 or K ∩ L = σ
for some σ ∈ E . Then we will denote σ = K|L .
(iii) For any K ∈ T , there exists a subset EK of E such that ∂K = K \K = ∪σ∈EKσ .
Furthermore, E = ∪K∈T EK and we will denote by N (K) the set of boundary volumes
of K that is N (K) = L ∈ T , K|L ∈ EK .
(iv) The family of points (xK)K∈T is such that xK ∈ K (for all K ∈ T ) and, if
σ = K|L , it is assumed that the straight line (xK , xL) is orthogonal to σ .
2.II.1. Numerical scheme 129
In the sequel we will use the following notation. The size of the mesh (or space step)
is defined by: size(T ) := maxK∈T δ(K) , where δ(K) denotes the diameter of the control
volume K . For any K ∈ T and σ ∈ E , m(K) is the d -dimensional Lebesgue measure
of K and m(σ) the (d−1) -dimensional Lebesgue measure of σ . The set of adjacent
couples (K,L) is denoted by Υ . The set of interior (resp. boundary) edges is denoted by
Eint (resp. Eext ), that is Eint = σ ∈ E ; σ 6⊂ ∂Ω (resp. Eext = σ ∈ E ; σ ⊂ ∂Ω ).
The set of external control volumes is denoted by Text . For all K ∈ T , L ∈ N (K) and
σ ∈ EK , we denote by xσ the orthogonal projection of xK on σ (thanks to assumption
(iv), this orthogonal projection is the same from xK or xL if σ = K|L ), by dK,L (resp.
dK,σ ) the Euclidean distance between xK and xL (resp. xσ ) and by νK (resp. νK,L
and νK,σ ) the outside unit normal to K (resp. with respect to L and with respect to σ ).
Remark that νK |σ = νK,σ for all σ ∈ EK and that thanks to assumption (iv), νK,L = νK,σ
if σ = K|L . Finally for all K ∈ T and σ ∈ EK we denote by S(K, σ) (resp. S(σ) ) the
”half-diamond” (resp. ”diamond”) associated to K and σ (resp. to σ ) that is the smallest
convex set that contains σ and xK (resp. S(σ) = S(K, σ) ∪ S(L, σ) ).Further, let T be an admissible mesh in the sense of Definition 2. Together with k ∈
(0, T ) , it generates the space-time grid ((n− 1)k, nk)×KK∈T ,n=1,...,[T/k]+1 , which we
denote by (T , k) . We define the grid step h by h := maxk, size(T ) . We denote by
QnK the space-time volumes, i.e., Qn
K = ((n − 1)k, nk) ×K for all n = 1, . . . , [T/k] + 1
and all K ∈ T . The lateral boundary of QnK is denoted by Σn
K . In addition, we introduce
the notation ν = 1/2minK,σ∈N (K) dK,σ and denote by CnK the cartesian product of some
fixed d -dimensional cube inside K , with edge 2ν/√d and center xK , and the interval
((n − 1)k, nk) . Furthermore, we will use the notation Υκ(QnK) for the union of all space-
time volumes of (T , k) that are separated from QnK by at most κ ∈ N (space or time)
interfaces.
Definition 3 (Admissible families of meches and grids) A family of meshes T is ad-
missible if each mesh is admissible and the following assumptions hold with some fixed M, ζ :
(v) there exists M such that
M ≥ maxK∈T
card(EK); (9)
(vi) there exists ζ > 0 such that
ζ size(T ) ≤ minK∈T ,σ∈EK
dK,σ. (10)
A family of space-time grids (T h, kh) , parametrized with h ∈ (0, 1) , is admissible if the
family T h of space meshes is admissible and maxkh, size(T h) ≤ h . In the sequel, we
write k for kh and (T h, k) for (T h, kh) .
130 Variational Approach for a Finite Volume Method
Remark 1 While Definition 2 and assumption (9) impose standard assumptions on the grid
(cf. [EyGaHe]), the hypothesis (10) is a strong proportionality condition. In fact, for p ≤ 2
it is only a technical assumption which permits to prove Lemma 5 (cf. Section 5). Lemma 5
is a pure qualitative result, in the sense that no estimation on the functions vh constructed
in this lemma is used in the sequel, but only the fact of their existence. Elsewhere, we can
replace (10) by the usual (cf. [EyGaHe]) assumption
ζ ≤ minK∈T h,σ∈EK
dK,σ
δ(K), (11)
which should be the only condition taken into account from the numerical point of view.
Nevertheless, in the case p > 2 the restriction (10) is essentially used in the proof of the
discret Poincare inequality (cf. the Appendix).
When using a finite volume method to approximate problem (1), we consider an ap-
proximate solution (if it exists) which is piecewise constant. Then we need to construct an
approximation of the gradient. Let us consider a set of values (vnK)K,n ⊂ RN (for the sake of
simplicity, we will make an abuse of notation by omitting K ∈ T h , n = 1, . . . , [T/k] + 1 )
and the dicrete solution defined by
vh|QnK= vnK . (12)
We construct the approximation of the gradient in the following way. The normal component
of the ”gradient” of the discrete solution vh is approximated by the operator Gh⊥ defined by
Gh⊥ : (vnK)K,n 7→ (gm⊥,σ)σ,m,
gm⊥,σ =
∣∣∣∣vmL − vmKdK,L
∣∣∣∣ ∈ [0,+∞) for σ = K|L,(13)
while the whole of the ”gradient” of the discrete solution vh on interfaces of the control
volumes is approximated by an operator Gh defined by
Gh : (vnK)K,n 7→ (gmσ )σ,m,
gmσ ∈ (Rd)N for all σ ∈ E , m = 1, . . . , [T/k] + 1.(14)
Then we extend Gh to the whole of Q by the lifting operator Lh defined by
Lh : (gmσ )σ,m 7→ Lh ((gmσ )σ,m) ∈ Lp(Q),
Lh ((gmσ )σ,m) |((m−1)k,mk)×S(σ) = gmσ for all σ ∈ E , m = 1, . . . , [T/k] + 1.(15)
We define the discrete gradient operator Dh by
Dh : (vnK)K,n 7→ (Lh Gh)((vnK)K,n) ∈ Lp(Q). (16)
2.II.1. Numerical scheme 131
It is convenient to extend the gradient approximation procedure to functions in E . Let
introduce the averaging operator Mh defined on E = Lp(0, T ;W 1,p(Ω))N by
Mh : η 7→ Mh η = (ηnK)K,n, ηnK =1
|CnK |
∫∫
CnK
η; (17)
Then by an abuse of notation, we also write Dh for the operator
Dh : η ∈ E 7→ (Lh Gh Mh)(η) ∈ Lp(Q), (18)
and Dh⊥ for the operator
Dh⊥ : η ∈ E 7→ (Lh Gh
⊥ Mh)(η) ∈ Lp(Q). (19)
Taking in (17) averages over cylinders CnK is convenient for expressing the consistency of the
gradient approximation for affine in x functions (cf. (iv) in the definition below). The crucial
property is the symmetry with respect to the axis x = xK .
Definition 4 (Admissible family of discrete gradient operators) For a given family of
meshes T h , a family of corresponding discrete gradient operators Dh is admissible, if
Dh and the corresponding operators Gh have the following properties:
(i) For each h , Gh is linear, i.e.,
Gh((vnK + wnK)K,n) = Gh((vnK)K,n) + Gh((wn
K)K,n) (20)
(ii) For each h , Gh is consistent with Gh⊥ , i.e.,
gmK|L νK,L = sign(vmL − vmK ) gm⊥,K|L for all (K|L) ∈ Υ, m = 1, . . . , [T/k] + 1. (21)
(iii) The family Dh is uniformly local, i.e., there exists κ ∈ N independent of h such that
for all K ∈ T h , all n = 1, . . . , [T/k] + 1 and all set of values (vnK)K,n of RN , there
exists a constant C which only depends on p , d and M , ζ in (R2) , (R3) such
that
∥∥Dh((vnK)K,n)∥∥Lp(Qn
K)≤ C
∥∥Dh⊥((v
nK)K,n)
∥∥Lp(Υκ(Qn
K)). (22)
(iv) For each h , Gh is consistent with affine functions. More exactly, assume that, for
K ∈ T h and n ∈ N given, there exists a constant c ∈ (Rd)N and a function w ∈ E
such that Dw ≡ c on Υκ(QnK) and that vlL = 1
|CnK |
∫∫CnKw whenever Ql
L ⊂ Υκ(QnK) .
Then
gnσ = c for all σ ∈ EK . (23)
132 Variational Approach for a Finite Volume Method
Below we provide an example of gradient approximation that complies with the properties
(i)-(iv) above. For simplicity, we restrict our attention to the 2D case.
Remark 2 (Example of admissible discrete gradients in 2D) Let T h be a family of
admissible meshes of Ω in the sense of Definition 2. For T ∈ T h , for each σ0 ∈ E , let
σ1 , σ2 , σ3 and σ4 be the four adjacent edges (see figure (1)). Let x1 (resp. x2 ) be
the intersection point of σ0 with σ1 and σ2 (resp. σ3 and σ4 ). Then we construct the
approximate gradient on σ0 in the following way.
We take the standard values gn⊥,σ0, gn⊥,σ1
of the normal components of the discrete
gradient on σ0 and σ1 , respectively. The edges are not colinear, so that there exists a
unique vector that has these values as projections on the normal directions to σ0 and σ1 ,
respectively. We draw this vector in x1 and denote it by gnσ01. In the same way, we can
reconstruct gnσ02(resp. gnσ03
, gnσ04) in x1 (resp. x2 ) from the normal components of the
gradients on σ0 and σ2 (resp. σ3 , σ4 ).
σ0
σ1
σ2
σ3
σ4
gn⊥,σ0
gn⊥,σ1
x1
x2
gnσ01
gnσ02
gnσ03
gnσ04
gnσ0
Figure 1: Reconstruction of gradient on σ0
Then the approximate gradient on σ0 (or in S(σ0) , according to (16)) is given by
gnσ0=
1
4(gnσ01
+ gnσ02+ gnσ03
+ gnσ04).
It is easy to show that, if the family T h is admissible in the sense of Definition 3, this
approximation of the gradient is admissible in the sense of Definition 4.
2.II.2. Existence of a discrete solution and a priori estimates 133
We are now in order to write the scheme. The equation for the scheme is given by
m(K)b(vnK)− b(vn−1
K )
k=
∑
L∈N (K)
m(K|L) ap(gnK|L) νK,L for all K ∈ T , n ∈ N. (24)
The initial condition is given by some values
u0K = b(v0K) for all K ∈ T , (25)
and the homogeneous Dirichlet boundary condition is taken into account in the following way:
vnK = 0 for all K ∈ Text, n ∈ N. (26)
The discrete Problem (Ph) corresponding to a grid (T h, k) , where T h is an admissible
mesh of Ω in the sense of Definition 2, is given by the discrete equation (24), the initial
condition (25) and the boundary condition (26). In Section 2 below, we prove that there
exists a solution vh to the discrete Problem (Ph) .
The values (u0K)K will be chosen in order to comply, at the limit h→ 0 , with the initial
condition in (1) and the restriction (4) on it, i.e.,u0,h → u0 in L1(Ω)N as h→ 0,
Ψ(u0,h) → Ψ(u0) in L1(Ω) as h→ 0,(27)
where u0,h|K = u0K . Our main result is the following theorem.
Theorem 1 (Convergence) Let (T hm , khm)m∈N be a sequence of admissible grids in the
sense of Definition 3 such that hm = maxkhm, size(T hm)
< 1 , hm → 0 as m→ +∞ .
Assume that (3),(4), and the analogue of (27) hold. For each m ∈ N , let (vhm)m∈N
be a discrete solution to the problem (Phm)m∈N , where (Dhm)m∈N is an admissible family
of discrete gradient operators in the sense of Definition 4. Then there exists a subsequence
(hmk)k∈N , hmk
→ 0 as k → ∞ , such that vhmk v in L1(Q) as hmk→ 0 , where
v ∈ E = Lp(0, T ;W 1,p0 (Ω))N is a weak solution of the problem (1) in the sense of Definition 1.
In the sequel, we will omit subscripts in sequences (hm) , (hmk) .
2 Existence of a discrete solution and a priori estimates
We will repeatedly use the following remark.
Remark 3 (Discrete integration by parts) Let T be an admissible mesh of Ω in the
sense of Definition 2. Let (vK)K∈T ⊂ RN and (FK,L)(K,L)∈Υ ⊂ R
N be real values such
that vK = 0 for all K ∈ Text and FK,L = −FL,K for all (K,L) ∈ Υ . Then
∑
K∈T
vK∑
L∈N (K)
FK,L =∑
(K,L)∈Υ
(vK − vL)FK,L. (28)
134 Variational Approach for a Finite Volume Method
Now we can state the result for existence of a discrete solution.
Theorem 2 (Existence) Let (T , k) be a space-time grid, where T is an admissible mesh
of Ω in the sense of Definition 2. Let Dh be a discrete gradient operator having the properties
(i)-(iii) of Definition 4. Assume that (3) holds; then there exists a solution vh to the discrete
problem (Ph) .
Remark 4 (Uniqueness) While uniqueness of a waek solution of the problem (1) itself for
N ≥ 2 seem to be an open problem, for special choices of the gradient approximation we can
also prove the uniqueness of a solution to the discrete problem (Ph) .
Proof of Theorem 2: Fix n ∈ 1, . . . , [T/k]+1 . Assume that the values (vn−1K )K∈T
are already found. We denote by V the vector of (RN)cardT whose entries (vnK)K∈T satisfy
the condition (26). Let us consider the operator S that associates to a “vector” V the
“vector” given by (24), i.e.,
S(V) =
m(K)
b(vnK)− b(vn−1K )
k−
∑
L∈N (K)
m(K|L) ap(gnK|L) · νK,L
K∈T
.
We are looking for a solution to the equation S(V) = 0 . Consider the scalar product
(S(V),V) in (RN)cardT . We have
1k
∑
K∈T
m(K) b(vnK) · vnK − 1k
∑
K∈T
m(K) b(vn−1K ) · vnK −
−∑
K∈T
∑
L∈N (K)
m(K|L) vnK ap(gnK|L)νK,L = 0.
(29)
In view of hypothesis (3), we have that the first term on the left-hand side of (29) is non-
negative. Since all the norms are equivalent on (RN)cardT , for the second term on the
left-hand side of (29) we have
1
k
∑
K∈T
m(K) b(vn−1K ) · vnK ≤ |V| 1
k
∑
K∈T
m(K) b(vn−1K ) = C |V| ;
here and in the sequel of the proof, C denotes a positive constant independent of V , and
|V| denotes the euclidean norm of V . We then handle the last term on the left-hand side of
(29), which we denote by AV . Using the discrete integration by parts (Remark 3), we obtain
that
−AV =∑
(K,L)∈Υ
m(K|L) (vnL − vnK) ap(gnK|L)νK,L.
In view of the definition (2) of ap and thanks to hypothesis (21), we obtain
−AV =∑
(K,L)∈Υ
m(K|L)∣∣gnK|L
∣∣p−2 (vnL − vnK)2
dK,L. (30)
2.II.2. Existence of a discrete solution and a priori estimates 135
For p ≥ 2 , we have in view of (21) that
∣∣gnK|L
∣∣p−2 ≥∣∣gn⊥,K|L
∣∣p−2=
∣∣∣∣vnL − vnKdK,L
∣∣∣∣p−2
,
which together with (30) yields
−AV ≥∑
(K,L)∈Υ
m(K|L) dK,L
∣∣∣∣vnL − vnKdK,L
∣∣∣∣p
.
Thanks to the discrete Poincare inequality (cf. Lemma 9 in the Appendix), we finally obtain
−AV ≥ α∑
K∈T
m(K) |vnK |p ≥ C |V|p ,
thanks to the equivalence of the norms on (RN)cardT .
For 1 < p ≤ 2 , we deduce from (20) that there exists C > 0 such that, for all
(K,L) ∈ Υ ,
C∣∣gnK|L
∣∣ ≤ |V| .
Together with (30), this yields
−AV ≥ C |V|p−2∑
(K,L)∈Υ
m(K|L) dK,L
∣∣∣∣vnL − vnKdK,L
∣∣∣∣2
.
Thanks to the discrete Poincare inequality (cf. Lemma 9 in the Appendix) for p = 2 , we
finally obtain
−AV ≥ C |V|p−2∑
K∈T
m(K) |vnK |2 ≥ c |V|p
with some constant c > 0 independent of V , thanks to the equivalence of the norms on
(RN)cardT .
Returning to (29), we obtain for all 1 < p <∞ that there exist constants c, C > 0 such
that
(S(V),V) ≥ c |V|p − C |V| ≥ 0,
for |V| large enough. Therefore in view of the Brouwer fixed point theorem (e.g., cf. [JLL,
Lemme 4.3]), there exists a solution to S(V) = 0 , i.e., there exists a solution to (24) with
condition (26) for (vn−1K )K given. ⋄
136 Variational Approach for a Finite Volume Method
Proposition 1 Let(T h, k)
be an admissible family of grids in the sense of Definition 3.
Let Dh be a family of discrete gradient operators satisfying (ii) and (iii) of Definition 4.
Assume that (3),(4), and (27) hold. Then for all family (vnK)K,n of solutions of the discrete
problem (Ph) there exists a constant C which only depends on p , Ω , T and ‖Ψ(u0)‖L1(Ω)
such that∫∫
Q
∣∣Dhvh∣∣p ≤ C, (31)
and
∑
K∈T h
m(K)B(vnK) ≤ C, for all n = 1, . . . , [T/k] + 1. (32)
Proof : Take i ∈ 1, . . . , [T/k] + 1 and multiply each term in (24) by viK . By (26),
using the discrete integration by parts (Remark 3) and (21), one gets
∑
K∈T h
m(K)(b(viK)− b(vi−1K )) · viK +
+k∑
(K,L)∈Υ
m(K|L)∣∣∣giK|L
∣∣∣p−2 ∣∣∣gi⊥,K|L
∣∣∣ |viK − viL| = 0.
By the convexity of Φ , one has (b(viK)−b(vi−1K )) ·viK ≥ B(viK)−B(vi−1
K ) . Summation over
i from 1 to n ∈ 1, . . . , [T/k]+1 , and recalling the definition of the discrete gradient and
its normal component, we infer
∑
K∈T h
m(K)B(vnK) + d
∫ nk
0
∫
Ω
∣∣∣Dhvh∣∣∣p−2∣∣∣Dh
⊥vh∣∣∣2
≤∑
K∈T h
m(K)Ψ(u0K). (33)
Note that, by (21), for 1 < p < 2 the value∣∣∣Dhvh
∣∣∣p−2∣∣∣Dh
⊥vh∣∣∣2
can be set zero whenever
Dh⊥v
h is zero. Therefore the integral in (33) always makes sense.
Now (32) follows directly from (27) and (33). As to (31), there are two cases. For p ≥ 2 ,
one has by (21) and (33)∫ ∫
Q
∣∣∣Dh⊥v
h∣∣∣p
≤∫ ∫
Q
∣∣∣Dhvh∣∣∣p−2∣∣∣Dh
⊥vh∣∣∣2
≤ const.
For 1 < p < 2 , using the “inverse” Holder inequality with the exponents p/2 < 1 and
p/(p− 2) < 0 , one gets
(∫ ∫
Q
∣∣∣Dh⊥v
h∣∣∣p)2/p
≤(∫ ∫
Q
∣∣∣Dhvh∣∣∣p)(2−p)/p ∫ ∫
Q
∣∣∣Dhvh∣∣∣p−2∣∣∣Dh
⊥vh∣∣∣2
.
In the two cases, it follows by (22) that∫∫
Q
∣∣∣Dh⊥v
h∣∣∣p
≤ const independently of h . Using
(22) again, one obtains (31). ⋄
2.II.3. Rewriting discrete equations under continuous form 137
3 Rewriting discrete equations under continuous form
Let vh be the discrete function (12) produced by the finite volume scheme (24)-(26) on a grid(T h, k
)of size h . We will replace the discrete gradient Dhvh by a function Dvh ∈ Lp(Q)
and the function b(vh) by a function uh ∈ L1(Q) so that (24) is equivalent to (8), i.e.,
uht = div ap(Dvh)
in D′(Q) . This representation plays the key role in proving the convergence result of Theo-
rem 1.
Define uh as the piecewise affine in t approximation of b(vh) :
uh(t, x)|QnK= b(vnK) +
t− kn
k(b(vnK)− b(vn−1
K )) (34)
Besides, for given K, n and a set(An
σ
)σ∈EK
⊂ (Rd)N , let AnK be defined a.e. on ∂K by
AnK |σ = An
σ for all σ ∈ EK . Let νK be the exterior unit normal vector to ∂K . Consider
the following Neumann problem in the factor space W =W 1,p(K)/R :
div ap(Dw) =1
m(K)
∫
∂K
AnKνK on K
ap(Dw)νK|∂K = AnKνK .
(35)
Lemma 1 Let AnK ∈ Lp′(∂K) . Then there exists a unique distribution solution wn
K to (35)
in W . This solution gives the global minimum to the functional
L : w ∈ W 7→ Lw =1
p‖Dw‖pLp(K) −
∫
∂K
wAnKνK +
1
m(K)
∫
K
w
∫
∂K
AnKνK . (36)
Proof : Supply W with the norm ‖w‖W =(∫
K|Dw|p
)1/p(we will ambiguously denote
by the same symbol an element of W 1,p(K) and the corresponding equivalence class as
element of W ). Consider the operator A : w ∈ W 7→ Aw ∈ W ′ defined by
< Aw,ϕ >W ′,W=
∫
K
ap(Dw) : Dϕ (37)
for all ϕ ∈ W , and the functional f ∈ W ′ defined by
< f, ϕ >W ′,W= − 1
m(K)
∫
K
ϕ
∫
∂K
AnKνK +
∫
∂K
ϕAnKνK (38)
for all ϕ ∈ W . Note that f is well defined, since the right-hand side of (38) is invariant
under translation by a constant in W 1,p(K) .
The operator A in (37) is bounded, hemicontinuous, strictly monotone on W , and
< Aw,w >W ′,W
‖w‖W=
∫K|Dw|p
‖w‖W= ‖w‖p−1
W → ∞
138 Variational Approach for a Finite Volume Method
as ‖w‖W → ∞ . Thus A is bijective (cf. e.g. [JLL, Chapitre 2,Theoreme 2.1]).
Besides, the functional L is well defined, convex on W , and
Lw ≥ 1
p‖Dw‖pLp(K) − ‖An
K‖Lp′(∂K)
∥∥∥∥w − 1
m(K)
∫
K
w
∥∥∥∥Lp(∂K)
≥
≥ 1
p‖Dw‖pLp(K) − const‖Dw‖Lp(K)
by Lemmae 12,11 (cf. the Appendix). Hence Lw → +∞ as ‖w‖W → ∞ , so that L
attains its global minimum on W (cf. e.g. [Br, Corollaire III.20]). Using the relation of the
p-laplacian with the Lp norm, by the standard variational argument we find out that this
minimum is attained at the unique solution wnK of the equation Aw = f in W ′ . ⋄
Let vnK be the solution of (35) with
AnK = ap(g
nσ) for σ ∈ EK , (39)
where gnσ are taken from (14), and vnK is normalized by assigning
1
m(K)
∫
K
vnK = vnK . (40)
We introduce the discrete-continuous approximation of Dhvh by setting
Dvh(t, x)|QnK= DvnK(t, x). (41)
Let us also define vh by
vh(t, x)|QnK= vnK(t, x). (42)
Clearly, we have uht = div ap(Dvh) pointwise on Qn
K . Note that Dvh is only the pointwise
gradient of vh , while the gradient of vh in the sense of distributions contains Dirac masses
concentrated on grid edges. Nevertheless, one has the following result.
Proposition 2 (The continuous form of (24)) Assume that (vnK)K,n verifies (24). Let
uh and Dvh be defined by (34) and (41), respectively. Then uht = div ap(Dvh) holds in
D′(Q) .
Proof : By the local conservativity of the scheme, the fluxes are continuous on the
space interfaces: AnK|L(x) = An
L|K(x) for all x ∈ K|L , for all (K,L) ∈ Υ , all n ∈
2.II.4. Proof of Theorem 1 139
1, . . . , [T/k] + 1 . Moreover, uh is continuous on the time interfaces. Therefore, for all
ph ∈ D(Q) ,
∫∫
Q
(uh · ϕt − ap(Dvh) : Dϕ) =
∑
K,n
∫∫
QnK
(uh · ϕt − ap(Dvh) : Dϕ) =
=∑
K,n
−∫∫
QnK
(uht − div ap(Dvh)) · ϕ+
∑
L∈N (K)
∫ kn
k(n−1)
∫
K|L
ϕAnK|LνK,L+
+
∫
K
ϕ(kn, ·) · uh(kn, ·)−∫
K
ϕ(k(n− 1), ·) · uh(k(n− 1), ·)=
=∑
n
∑
(K,L)∈Υ
∫ kn
k(n−1)
∫
K|L
ϕAnK|L(νK,L + νL,K)+
+
∫
K
ϕ(T, ·) · uh(T, ·)−∫
K
ϕ(0, ·) · uh(0, ·) = 0,
so that (8) holds. ⋄
In addition to this result, it will be useful to have in hand some “Lp(0, T ;W 1,p0 (Ω)) -
version” of vh for each h . We will call vh a family of continuous in x approximations
of vh , if
vh ∈ E with ‖vh‖E ≤ const‖Dhvh‖Lp(Q) (43)
with a constant independent of h ,
‖vh − vh‖L1(Q) → 0 as h→ 0, (44)
and
1
km(K)
∫∫
QnK
vh = vnK =1
|CnK |
∫∫
CnK
vh. (45)
The existence of such approximations is proved in Lemma 5 (cf. Section 5) for the case where
(10) holds.
4 The proof of Theorem 1
In Chapter 2.I, in the context of continuous dependence upon the data of weak solutions to
“general” elliptic-parabolic problems, the convergence proof for weak solutions of approximat-
ing problems has been reduced to the three essential arguments:
(A) a priori estimates, by the chain rule argument of Alt-Luckhaus (cf. [AL83, Ot96,
CaW99, BAB]);
(B) strong compactness in the parabolic term, by the Kruzhkov lemma (cf. [K69a] and
Chapter 2.I, Lemma 6);
140 Variational Approach for a Finite Volume Method
(C) convergence in the elliptic term, by the Minty-Browder argument (cf. e.g. [Mi62, Mi63,
Bro63, JLL] and Chapter 2.I, Lemma 7).
Here we will take advantage of the “continuous” form (8) of the system (24) and pass to the
limit in (a subsequence of) vh as h→ 0 by applying the same arguments to, respectively,
(A) vh , the continuous in x approximations of vh ;
(B) uh , the piecewise affine in t approximations of b(vh) ;
(C) Dvh , the discrete-continuous approximations of Dhvh .
Proof of Theorem 1: We will repeatedly refer to results contained in Sections 5-7
below. The proof consists of three steps.
(A) Let vh be a family of solutions to the family of discrete problems (Ph) . By Lemma 5
(cf. Section 5) there exist vh ∈ E satisfying (43)-(45). In particular, by Proposition 1,
‖vh‖E ≤ const uniformly in h . Hence there exists a subsequence h → 0 and a function
v ∈ E such that vh v in E as h → 0 . By (44), one also has vh v ∈ L1(Q) (cf.
also Remark 5).
(B) We claim that the family uh is relatively compact in L1(Q) . Indeed, extend uh
by zero on (R× Rd) \Q . Let us check the following three conditions:
(i) uh is bounded in L1(Q) ;
(ii) ap(Dvh) is bounded in L1(Q) ;
(iii) for all ∆ > 0 small enough, one has
sup|∆x|≤∆
∫∫
Q
|uh(t, x+∆x)− uh(t, x)| dxdt ≤ ωx(∆) (46)
uniformly in h , where ωx(∆) → 0 as ∆ → 0 .
In order to prove (i), note that
|b(z)| ≤ δB(z) + sup|ζ|≤1/δ
|b(ζ)| (47)
holds for all δ > 0 (cf. e.g. [AL83]). Hence we have by (34),(25)
‖uh‖L1(Q) ≤ 2
∫∫
Q
|b(vh)|+ k∑
K∈T h
m(K)|u0K| ≤∫∫
Q
B(vh) + const + k∑
K∈T h
m(K)|u0K |,
which is bounded uniformly in h , by Proposition 1 and (27).
As to (ii), one has
‖ap(Dvh)‖L1(Q) =
∫∫
Q
|Dvh|p−1 ≤ ‖Dvh‖p−1Lp(Q)|Q|1/p ≤ const
2.II.4. Proof of Theorem 1 141
by the a priori estimate on ‖Dvh‖Lp(Q) , which is proved in Lemma 2 (cf. Section 5).
The estimate (46) of (iii) is proved in Lemma 6, as a consequence of the a priori estimate
(31) (cf. Section 6).
Now we may conclude by Lemma 6 from Chapter 2.I that
sup|∆t|≤∆
∫∫
Q
|uh(t+∆t, x)− uh(t, x)| dxdt ≤ ωt(∆) (48)
uniformly in h , where ωt(∆) → 0 as ∆ → 0 . Thus there exists a subsequence h→ 0 and
a function u ∈ L1(Q) such that uh → u in L1(Q) and a.e. on Q . Besides, we have to
establish that u = b(v) , where v is the weak limit of vh in E . The proof, which follows
the idea of [BrSt73], is given in Lemma 7 (cf. Section 6).
(C) First note that uh → b(v) in L1(Q) , so that uht → b(v)t in D′(Q) . Moreover, by
(8) ‖uht ‖E′ = ‖ap(Dvh)‖Lp′(Q) = ‖Dvh‖p−1Lp(Q) , which is bounded by Lemma 2 (cf. Section 5).
Therefore uht is weak- ∗ relatively compact in E ′ .
It follows that there exists a subsequence, which we abusively denote by h→ 0 , such that
(i) vh v in E ;
(ii) −uht∗ −b(v)t in E ′ .
Moreover, for all h one has
(iii) uht = Ahvh , where vh is constructed in Lemma 5 and Ah is the operator that maps
η ∈ E to −divAhη ∈ E ′ , with Ah : E 7→ Lp′(Q) defined in (51) below.
Indeed, let us define the finite volume approximate Ah· of ap(D·) . For η ∈ E and all
space-time volume QnK , set
Dηh(t, x)|QnK= DηnK(x), (49)
where ηnK is the unique solution to the problem (35) with AnK |σ = ap
(((Gh Mh)η)nσ
)for
all σ ∈ EK . It is convenient to normalize ηnK by assigning
1
m(K)
∫
K
ηnK =1
km(K)
∫∫
QnK
η. (50)
Here Mh and Gh are the averaging operator and the gradient approximation operator,
respectively, defined by (17) and (14), respectively. Assign
Ahη = ap(Dηh). (51)
From (45) we have(Gh Mh
)vh = Gh
((vnK)K,n
). Therefore, Ahvh = ap(Dv
h) with Dvh
defined in (41), so that (8) yields (iii).
142 Variational Approach for a Finite Volume Method
Arguing as in the proof of Proposition 2, one gets
< Ahη, ϕ >E′,E=
∫∫
Q
ap(Dηh) : Dϕ =
∫∫
Q
Ahη : Dϕ (52)
for all ϕ ∈ E . In particular, it follows that Ah is monotone for all h .
Now we apply Lemma 7 from Chapter 2.I to Ah and A : E 7→ E ′ defined by Aη =
−div ap(Dη) , or, equivalently, by
< Aη, ϕ >E′,E=
∫∫
Q
ap(Dη) : Dϕ (53)
for all ϕ ∈ E . Note that A is hemicontinuous. Two more assumptions of Lemma 7
(Chapter 2.I) have to be checked:
(iv) lim infh→0 < −uht , vh >E′,E ≤ < −b(v)t, v >E′,E ;
(v) for all η ∈ E , Ahη → Aη in E ′ .
In fact the property (v) expresses the consistency in E ′ of the finite volume approximation
of the operator −div ap(D·) on the space E . We prove in Theorem 3 (cf. Section 7) that
(v) actually holds under the assumptions of Theorem 1.
Before proving (iv), let us show that the initial condition (6) holds. From (34) we have
< −uht , ζ >E′,E= −∫∫
Q
uht · ζ =∑
K∈T h
m(K)u0K · ζ(0, ·) +∫∫
Q
uh · ζt (54)
for all ζ ∈ E with ζt ∈ L∞(Q) and ζ(T, ·) = 0 . By (27), we can pass to the limit in (54)
and obtain (6). Now we can apply the usual chain rule argument (cf. [AL83, Lemma 1.5])
and deduce that B(v) ∈ L∞(0, T ;L1(Ω)) and
< −b(v)t, v >E′,E= −∫
Ω
B(v)(T ) +
∫
Ω
Ψ(u0). (55)
(without loss of generality, we assume T to be a Lebesgue point of ‖B(v)‖L1(Ω)(·) ). Besides,by (8),(34),(45), and the monotonicity of b(·) , one has
< −uht , vh >E′,E= −1
k
∑
K,n
(b(vnK)− b(vn−1K )) ·
∫∫
QnK
vh =
=∑
K∈T h
[T/k]+1∑
n=1
m(K)(b(vnK)− b(vn−1K )) · vnK ≤
−∑
K∈T h
m(K)B(v[T/k]+1K ) +
∑
K∈T h
m(K)Ψ(u0K).
Together with (55),(27) and the Fatou lemma, this yields (iv).
We are now in position to conclude that −b(v)t = Av in E ′ , so that (5) also holds.
Thus v is a solution of (1) in the sense of Definition 1. ⋄
Remark 5 In fact, one could replace in the requirement (44) the space L1(Q) by the space
Lp(Q) , and show that vh v in Lp(Q) .
2.II.5. Two kinds of continuous approximations 143
5 Two kinds of continuous approximations
In this section we prove two auxiliary results concerning the approximations Dvh and vh of
Dhvh and vh , respectively. We also establish a uniform estimate on the space translates of
vh in Lq(Q) , 1 ≤ q ≤ p .
Lemma 2 Let(T h, k)
be an admissible family of grids, and (vnK)K,n be a solution of
(Ph) . Assume that the family of discrete gradient operators Dh satisfy (ii),(iii) of Defini-
tion 4, and let Dvh be defined by (41). Then ‖Dvh‖Lp(Q) ≤ const uniformly in h .
Proof : For all K, n fixed, the function vnK ∈ W 1,p(K) satisfies (35) in D′(K) . Take
vnK for the test function; it follows by (39) that∫
K
|DvnK |p =∑
σ∈EK
m(σ)( 1
m(σ)
∫
σ
vnK − 1
m(K)
∫
K
vnK
)(ap(g
nσ)νK,σ
).
Multiplying and dividing each term of the sum in the right-hand side by dK,σ/d , integrating
in t over (k(n− 1), kn) and summing over K, n , we obtain by the Holder inequality
‖Dvh‖pLp(Q) ≤ d(∑
K,n
∑
σ∈EK
1
dkm(K)dK,σ|ap(gnσ)|p
′)1/p′
×
×(∑
K,n
∑
σ∈EK
1
dkm(K)dK,σ
∣∣∣∣∣
1m(σ)
∫σvnK − 1
m(K)
∫KvnK
dK,σ
∣∣∣∣∣
p)1/p.
(56)
According to (18),(15) and (45), the first term in the right-hand side of (56) equals(∫∫
Q
(|Dhvh|p−1)p′)1/p′
= ‖Dhvh‖p−1Lp(Q) = ‖Dhvh‖p−1
Lp(Q),
which is bounded by Proposition 1. Besides, by Lemma 10 (cf. the Appendix) the second
term in the right-hand side of (56) is estimated by the value
const(∑
K,n
‖DvnK‖pLp(QnK)
)1/p= const‖DvnK‖Lp(Q).
Hence (56) yields
‖Dvh‖pLp(Q) ≤ const‖Dvh‖Lp(Q)
with a constant independent of h , which completes the proof. ⋄
Lemma 3 Let(T h, k)
be an admissible family of grids, (vnK)K,n be a solution of (Ph) ,
and vh be defined by (12) on Q and extended by zero on (R+×R) \Q . For x,∆x ∈ Rd ,
let Bx,x+∆x be the broken line that joins the centers of successive space mesh volumes crossed
by [x, x+∆x] ∩ Ω , and lx,x+∆x be the length of Bx,x+∆x ∩ Ω . Then for 1 ≤ q ≤ p ,∫∫
Q
|vh(t, x+∆x)− vh(t, x)|q dxdt ≤ ω(q)x (|∆x|) (57)
uniformly in h , with ω(q)x : R+ 7→ R
+ such that ω(q)x (∆) → 0 as ∆ → 0 . Moreover, one
can take ω(q)x (∆) = const∆(l(∆))q−1 , where l(∆) = supx∈Rd sup∆x∈Rd,|∆x|≤∆ lx,x+∆x .
144 Variational Approach for a Finite Volume Method
Proof : Take x,∆x ∈ Rd and let Bx,x+∆x =
⋃(K,L)∈Υ | [x,x+∆x]∩(K|L)6=Ø[xK , xL] , where
xK , xL are the centers of the volumes K,L , respectively. Since
lx,x+∆x =∑
(K,L)∈Υ | [x,x+∆x]∩(K|L)6=Ø
dK,L,
convexity of the mapping y ∈ Rd 7→ |y|q yields
|vh(t, x+∆x)− vh(t, x)|q ≤ (lx,x+∆x)q−1
∑
(K,L)∈Υ | [x,x+∆x]∩(K|L)6=Ø
dK,L
∣∣∣∣vnK − vnLdK,L
∣∣∣∣q
for t ∈ (k(n − 1), kn) . For (K,L) ∈ Υ fixed, denote by ΩKL(∆x) the set of x ∈ Rd
such that [x, x+∆x] ∩ (K|L) 6= Ø . Clearly, ΩKL(∆x) is a prism of measure less or equal
than |∆x|m(K|L) . Hence∫∫
Q
|vh(t, x+∆x)− vh(t, x)|q ≤ (l(|∆x|))q−1∑
K,n
kdK,L
∣∣∣∣vnK − vnLdK,L
∣∣∣∣q
|ΩKL(∆x)| ≤
≤ d|∆x|(l(|∆x|))q−1
∫∫
Q
|Dh⊥v
h|q ≤ const|∆x|(l(|∆x|))q−1
by Proposition 1 and the Holder inequality. ⋄
For q > 1 , this result will be used together with the simple lemma below.
Lemma 4 Let T h be a family of admissible meshes of Ω ⊂ Rd parametrized by h =
size(T h) such that (10) holds. Let x,∆x ∈ Rd and Bx,x+∆x , lx,x+∆x , l(·) be defined as
in Lemma 3. Then l(∆) ≤ const (∆ + 2h) , where const only depends on d and on ζ in
(10).
Proof : Let Cx,x+∆x be the cylinder in Rd of radius h , with the segment [x−h ∆x
|∆x| , x+
∆x + h ∆x|∆x| ] for the axis. If the segment [x, x + ∆x] crosses a control volume K ∈ T h ,
K is contained in Cx,x+∆x . Note that |Cx,x+∆x| = const (|∆x| + 2h) hd−1 . On the other
hand, for all K ∈ T h , |K| ≥ const (ζh)d , and the maximum length of Bx,x+∆x ∩ K is
2h . Therefore l(|∆x|) is estimated by 2h times the number of control volumes contained
in Cx,x+∆x , i.e., l(|∆x|) ≤ 2h const (|∆x|+2h)hd−1
consthd , which concludes the proof. ⋄
Remark 6 In fact, we prove in Lemma 3 that for any mesh T admissible in the sense of
Definition 2, for all (wK)K ⊂ RN given, and w defined by w|K = wK , the norm of
w(·+∆x)−w(·) in Lq(Ω) is estimated by dC |∆x|(l(|∆x|))q−1 , where C is the discrete
W 1,q0 -norm of w . As it is shown in Lemma 4, for a family T h this estimate can be
improved to const dC |∆x|(|∆x| + 2h)q−1 , with const independent of h = size(T h) ,
provided one imposes the restriction of proportionality (10) on T h .
Moreover, calculating more carefully |ΩKL(∆x)| (cf. e.g. [EyGaHe, EGH98]), we can
prove this last estimate in case 1 ≤ q ≤ 2 without the restriction (10).
2.II.5. Two kinds of continuous approximations 145
Lemma 5 Let(T h, k)
be an admissible family of grids, (vnK)K,n be a solution of (Ph) ,
and vh be defined by (12). Then there exists a family vh ⊂ E such that (43)-(45) hold.
Proof : We first convolute vh in x with a special mollifier, and then restore the average
over each mesh volume.
Let ν = 1/2minK∈T h,σ∈EK dK,σ . By (10) we have
h/ν ≤ const. (58)
Take ρ : x ∈ Rd 7→ d
|B(0,1)|dist (x, S(0, 1)) , where S(0, r) , B(0, r) are the (d− 1) -
dimensional sphere and the d -dimensional ball of centre 0 and radius r , respectively. Let
Ων = x ∈ Ω | dist (x, ∂Ω) > ν and χΩν be the characteristic function of Ων . We set
vh,ν =1
νdρ(xν
)∗ (vh χΩν ) (59)
and construct vh as
vh = vh,ν +∑
K∈T h
αKϕK . (60)
Here ϕK(t, x) = ϕK(x) = m(K)νd
π(x−xK
ν
). Here π : R
d 7→ R is a function with the
properties supp π = x ∈ Rd | 1 ≤ |x| ≤ 2 , π ≥ 0 ,
∫π = 1 , π ∈ C∞(Rd) ; xK is the
center of K , and
αK(t, x)|QnK= αn
K = vnK − 1
km(K)
∫∫
QnK
vh,ν. (61)
Note that ϕK ∈ E ; by (10) and (58) |ϕK | ≤ const and |DϕK | ≤ const/h with const
independent of h . Moreover, 1km(K)
∫∫Qn
KϕK = 1 . In addition, by the choice of ν each
control volume K ∈ T h contains a ball of radius 2ν . Hence, by (59),(60) and the definition
of π , vh|CnK≡ vh,ν |Cn
K= vnK . Therefore vh verify both equalities in (45).
By Lemma 3, (vh(·+∆x, ·)−vh(·, ·)) vanish in L1(Q) uniformly in h and ∆x ∈ B(0, ν)
as ν goes to zero. A fortiori, the same holds with the function vh replaced by vh χΩν .
Since ν → 0 as h → 0 , by the usual property of convolution regularisations (cf. e.g. [Br,
Theoreme IV.22]), ‖vh,ν − vh χΩν‖L1(Q) → 0 as h→ 0 . Moreover, it follows by the Holder
inequality, the discrete Poincare inequality (cf. Lemma 9 in the Appendix), and Proposition 1
that∫ T
0
∫
Ω\Ων
|vh| ≤(∫∫
Q
|vh|p)1/p (
T |Ω \ Ων |)1/p′
≤ const(|Ω \ Ων |
)1/p′→ 0
as h→ 0 . Therefore
‖vh,ν − vh‖L1(Q) → 0 (62)
146 Variational Approach for a Finite Volume Method
as h→ 0 . Finally,
‖vh − vh,ν‖L1(Q) =∑
K,n
|αnK |∫∫
QnK
ϕK =
=∑
K,n
1
km(K)
∣∣∣∣∣
∫∫
QnK
vh −∫∫
QnK
vh,ν
∣∣∣∣∣
∫∫
QnK
ϕK ≤
≤ const∑
K,n
∫∫
QnK
|vh − vh,ν| = const ‖vh − vh,ν‖L1(Q) → 0
as h→ 0 . Therefore (44) holds.
It remains to prove (43). First, let us estimate vh,ν in E . It is convenient to write
Dρ(·) in the spherical coordinates; indeed, if |x| = r and x/|x| = eθ ∈ S(0, 1) , we
have Dρ(x) = const eθχB(0,1) a.e. on Rd . Let S+(0, r) = x ∈ S(0, r) | x1 > 0 and
B+(0, r) = x ∈ B(0, r) | x1 > 0 . Denote vh χΩν by f . Separating the two hemispheres
S+(0, 1) and S(0, 1) \ S+(0, 1) , we find
|Dvh,ν(t, x)| = 1
νd+1
∣∣∣∣∫Dρ(x− y
ν
)f(t, y) dy
∣∣∣∣ =
=1
νd+1
∣∣∣∣∫
B(0,ν)
Dρ(σν
)f(t, x− σ) dσ
∣∣∣∣ =
=const
νd+1
∣∣∣∣∫
S(0,1)
eθ
∫ ν
0
f(t, x− reθ)rd−1 drdθ
∣∣∣∣ =
=const
νd+1
∣∣∣∣∫
S+(0,1)
∫ ν
0
|f(t, x+ reθ)− f(t, x− reθ)|rd−1 drdθ
∣∣∣∣ ≤
≤ const
νd+1
∫∫
B+(0,ν)
∫ ν
0
|f(t, x+ σ)− f(t, x− σ)| dσ.
Therefore by the Holder inequality one has
∫∫
Q
|Dvh,ν|p ≤ const
νp(d+1)
∫∫
Q
∫
B+(0,ν)
|f(t, x+ σ)− f(t, x− σ)|p(const νd)p/p′ dσdxdt ≤
≤ const ν−d−p
∫
B+(0,ν)
∫∫
Q
|vh(t, x+ σ)− vh(t, x)|p dxdtdσ ≤
≤ const ν−p sup|σ|≤ν
∫∫
Q
|vh(t, x+ σ)− vh(t, x)|p dxdt.
Hence by Lemma 3, (58) and Lemma 4 we finally deduce that ‖Dvh,ν‖Lp(Q) ≤ const uni-
formly in h .
Now we are able to estimate (vh − vh,ν) in E . In fact, we have
∫∫
Q
|D(vh − vh,ν)|p =∑
K,n
|αnK |p∫∫
QnK
|DϕK|p ≤const
hp
∑
K,n
km(K)|αnK |p. (63)
2.II.6. Compactness result for the parabolic term 147
Moreover,
|αnK |p =
1
(km(K))p
∣∣∣∣∣
∫∫
QnK
(vh,ν − vh)
∣∣∣∣∣
p
≤ 1
(km(K))p(km(K))p/p
′×
×∫∫
QnK
∣∣vh,ν − vh∣∣p ≤ 1
km(K)const δ(K)p
∫∫
QnK
|Dvh,ν|p,(64)
since Dvh ≡ 0 on QnK . The inequality in (64) holds, because, as we have already seen,
vh,ν|CnK
≡ vnK ≡ vh|CnK, and the Poincare inequality for (vh,ν − vh) in K therefore holds
with const independent of h, ν . Substituting (64) into (63) and using (58), we obtain the
desired estimate and prove (43). ⋄
6 The compactness result for the parabolic term
In this section we prove the uniform estimate (46) on space translates of uh in L1(Q) ,
starting from the result of Lemma 3 in Section 5. Then we identify the function u = limh→0 uh
in L1(Q) (for a subsequence) with b(v) , where v = weak− limh→0 vh in E .
Lemma 6 Let(T h, k)
be an admissible family of grids, (vnK)K,n be a solution of (Ph) ,
and uh be defined by (34). Then (46) holds.
Proof : It is a modification of the corresponding proofs in [AL83] and Chapter 2.I.
First note that∫∫
Q
|uh(t, x+∆x)− uh(t, x)| dxdt ≤ k
∫
Ω
|u0,h(x+∆x)− u0,h(x)| dx+
+2
∫∫
Q
|b(vh(t, x+∆x))− b(vh(t, x))|.(65)
By (27), the set u0,h is compact in L1(Q) , so that
sup|∆x|≤∆
∫
Ω
|u0,h(x+∆x)− u0,h(x)| dx ≤ ω0(∆)
with ω0 : R+ 7→ R+ such that ω0(∆) → 0 as ∆ → 0 , uniformly in h . Moreover, note that
b(vh) are equiintegrable on Q . Indeed, for all set F ⊂ Q one has by (47) and Proposition 1∫∫
F
|b(vh)| ≤ infδ>0
(δ
∫∫
Q
B(vh) + C(δ)|F |)≤ inf
δ>0
(const δ + C(δ)|F |
)= ω1(|F |),
and ω1(|F |) → 0 as |F | → 0 .
Further, for M > 0 let us introduce RhM = (t, x) ∈ Q| |vh(t, x)| ≤M, |vh(t, x+∆x)| ≤
M . It follows from Proposition 1 and the discrete Poincare inequality (Lemma 9 in the
Appendix) that ‖vh‖L1(Q) ≤ const uniformly in h . Hence |Q \ RhM | → 0 as M → +∞
uniformly in h , by the Chebyshev inequality. Let ωb,M(·) be the modulus of continuity of
148 Variational Approach for a Finite Volume Method
b(·) on [−M,M ]N . Integrating separately over Q \RhM and Rh
M in the last term in (65),
we get∫∫
Q
|uh(t, x+∆x)− uh(t, x)| ≤ ω0(|∆x|) + 2ω1(|Q \RhM |)+
+
∫∫
RhM
ωb,M(|vh(t, x+∆x)− vh(t, x)|) dxdt.
It follows by the concavity of ωb,M(·) and Lemma 3 that
sup|∆x|≤∆
∫∫
Q
|uh(t, x+∆x)− uh(t, x)| dxdt ≤ infM>0,δ>0
ω0(|∆x|) + 2ω1(sup
h|Q \Rh
M |)+
+|Q|ωb,M
( 1
|Q|
∫∫
Q
|vh(t, x+∆x)− vh(t, x)| dxdt)
= ωx(∆),
and ωx(∆) → 0 as ∆ → 0 uniformly in h . ⋄
Lemma 7 Let(T h, k)
be an admissible family of grids and (vnK)K,n be a solution of
(Ph) . Assume that vh v in E and uh → u in L1(Q) for a sequence h → 0 , where
uh are defined by (34) and vh satisfy (43),(44). Then u = b(v) .
Proof : We claim that vh v in L1(Q) and b(vh) → u in L1(Q) , and then apply
the usual monotonicity argument (cf. [BrSt73]).
Since vh v also in L1(Q) , the first claim follows from (44). Further, let us show that
uh → u in L1(Q) implies ‖uh − b(vh)‖L1(Q) → 0 as h→ 0 . We have
‖uh − b(vh)‖L1(Q) =∑
K,n
∫∫
QnK
∣∣∣∣b(vnK) +
t− kn
k(b(vnK)− b(vn−1
K ))
∣∣∣∣ dxdt =
=1
2
∑
K,n
km(K)|b(vnK)− b(vn−1K )| ≤ 2
∑
K,n
∫∫
QnK
|uh(t + k, x)− uh(t, x)| dxdt.(66)
The last inequality follows form the “geometrical” observation that
k|β − α|+ k|γ − β| ≤ 4
∫ k
0
|α+ θ(β − α)− β − θ(γ − β)| dθfor all k > 0, α, β, γ ∈ R.
(67)
Indeed, it is easily checked that for L = |β−α|+|γ−β| fixed, the minimum of the right-hand
side of (67) is attained at (β−α) = −(γ − β) and equals 14kL . Recall (34); it is sufficient
to apply (67) to α = b(vn−1K ) , β = b(vnK) , γ = b(vn+1
K ) in order to obtain (66). Note that,
by the compactness of uh in L1(Q) , (48) holds. Therefore the right-hand side of (66)
vanishes as h→ 0 .
Without loss of generality, we can assume that b(vh) → u a.e. on Q . For each ε > 0 ,
choose a set Rε ⊂ Q , with |Rε| < ε , such that v ∈ L∞(Q \ Rε) and b(vh) → u in
L∞(Q \Rε) . This is always possible, by the Chebyshev inequality and the Egorov Theorem.
2.II.7. Consistency of the finite volume approximation 149
Consequently, we have b(v + λζ) → b(v) in L∞(Q \ Rε) as λ ∈ R tends to zero, for all
ζ ∈ L∞(Q) . Then for all η ∈ L∞(Q)
∫∫
Q\Rε
u · (v − η) = limh→0
∫∫
Q\Rε
b(vh) · (vh − η) ≥
≥ limh→0
∫∫
Q\Rε
b(η) · (vh − η) =
∫∫
Q\Rε
b(η) · (v − η).(68)
The inequality in (68) is due to the monotonicity of b(·) :∫∫Q\Rε
(b(vh)−b(η)) · (vh−η) ≥ 0 .
Now it is sufficient to take η = v + λζ with λ ↑ 0 and λ ↓ 0 in order to deduce that
±∫∫
Q\Rε(u − b(v)) · ζ ≥ 0 . Since ζ ∈ L∞(Q) and ε > 0 are arbitrary, u = b(v) a.e. on
Q . ⋄
7 Consistency of the finite volume approximation
In this section we prove that, for an arbitrary function η ∈ E , the finite volume approximation
Ahη = −divAhη defined by (52),(49) is a good approximation for Aη = div ap(Dη) in E ′ .
More exactly, we have the following definition.
Definition 5 (Consistent approximations) Let (T h) be a family of admissible meshes,
parametrized by h→ 0 ,(T h, kh)
be the corresponding grids with maxsize(T h), kh ≤
h , and Dh be a family of discrete gradient operators. Let Ah : E 7→ E ′ be the operator
defined by (52),(49).
We say that Ah is the approximation of the elliptic operator A · = −div ap(D ·) corre-
sponding to(T h, kh)
and Dh . This approximation is consistent if for all η ∈ E one
has Ahη → Aη in E ′ as h→ 0 .
7.1 Properties of finite volume approximations
of the elliptic term and the consistency
Theorem 3 (Consistency) Let the family of grids and the family of discrete gradient oper-
ators be admissible in the sense of Definitions 3 and 4, respectively. Then the corresponding
approximation of −div ap(D ·) is consistent.
The main ingredient of the proof is the following result. Recall that Υκ(QnK) denotes
the union of all space-time volumes of (T h, k) that are separated from QnK by at most κ
(space or time) interfaces.
Proposition 3 Let(T h, k)
be an admissible family of grids, and Dh be an admissible
family of discrete gradient operators. Then the operators Ah : E 7→ Lp′(Q) defined by
(51),(49) have the following properties:
150 Variational Approach for a Finite Volume Method
(i) The operators Ah are uniformly local, i.e., there exists a constant C , independent of
h , such that for all η ∈ E , for all set H ⊂ Q such that H =⋃m
i=1QniKi
, one has
‖Ahη‖p′Lp′(H)
= ‖Dηh‖pLp(H) ≤ C ‖Dη‖pLp(Υκ+1(H)),
where Υκ+1(H) =⋃m
i=1Υκ+1(QniKi) .
(ii) The operators Ah are locally Holder equi-continuous, i.e., for all R > 0 there exists a
constant C(R) , independent of h , such that
‖Ahη −Ahµ‖Lp′(Q) ≤ C(R)‖η − µ‖αE
whenever ‖η‖E ≤ R , ‖µ‖E ≤ R . Here α = 1/2 for p ≥ 2 and α = p/(p′)2 for
1 < p ≤ 2 .
Proof of Theorem 3: We have to prove that ‖ap(Dη)−Ahη‖Lp′(Q) → 0 as h→ 0 .
Let us first prove the theorem for the case of η ∈ E that is piecewise constant in t and
piecewise affine in x . Let J ⊂ Q be the set of discontinuities of Dη . Clearly, J is of finite
d -dimensional Hausdorff measure Hd(J) . Let us introduce Hh =⋃
K,n |Υκ(QnK)∩J 6=ØQ
nK .
Note that |Hh| ≤ (κ + 1)hHd(J) → 0 as h → 0 ; likewise, |Υκ+1(Hh)| → 0 as h → 0 .
Therefore by (2) and Proposition 3(i) we have
∫∫
Hh
|ap(Dη)− ap(Dηh)|p′ ≤
∫∫
Hh
|Dη|p + C
∫∫
Υκ+1(Hh)
|Dη|p → 0
as h → 0 . Besides, for all QnK such that Qn
K ∩ Hh = Ø we have Dηh ≡ Dη on QnK .
Indeed, we have Dη ≡ const on Υκ+1(QnK) . Therefore Dhη|Qn
K≡ Dη = const by (23).
Hence Dw = Dη satisfies the boundary condition in (35); the equation is also satisfied, since
div ap(Dhη) ≡ 0 on QnK and 1
km(K)
∫∫Σn
Kap(Dhη)νK = ap(Dη)
∫∂KνK = 0 .
It follows that
‖ap(Dη)−Ahη‖Lp′ (Q) =(∫∫
Hh
|ap(Dη)− ap(Dηh)|p′
)1/p′→ 0
as h→ 0 , which was our claim.
Now let us approximate an arbitrary function η in E by functions µl that are piecewise
constant in t and piecewise affine in x . More exactly, there exists a sequence µl in E such
that µl → η in E and a.e. on Q as l → ∞ , and |Dµl|p are dominated by an L1(Q)
function independent of l . We have
‖ap(Dη)−Ahη‖Lp′(Q) ≤ ‖ap(Dη)− ap(Dµl)‖Lp′ (Q)+
+‖ap(Dµl)−Ahµl‖Lp′(Q) + ‖Ahµl −Ahη‖Lp′(Q).(69)
2.II.7. Consistency of the finite volume approximation 151
As l → 0 , the first term in the right-hand side of (69) converges to zero by the Lebesgue
dominated convergence theorem, independently of h . The second one converges to zero as
h → 0 , for all l fixed. Finally, by Proposition 3(i) and (ii), the third one converges to zero
as l → ∞ , uniformly in h . Hence the left-hand side of (69) can be made as small as desired
for h sufficiently small. This concludes the proof. ⋄
7.2 Proof of Proposition 3
The main ingredient is the following lemma.
Lemma 8 Let(T h, k)
be an admissible family of grids, and Dh be an admissible
family of discrete gradient operators. Let η, µ ∈ E and H ⊂ Q such that H =⋃m
i=1QniKi
.
Then
(i) for all R > 0 there exists a constant C(R) , independent of h , such that
m∑
i=1
‖ap(Dhη)− ap(Dhµ)‖p′Lp′(Σ
niKi
)≤ C(R)
h‖Dη −Dµ‖minp,p′
Lp(Υκ+1(H))
whenever ‖η‖E ≤ R , ‖µ‖E ≤ R ; here Υκ+1(H) =⋃m
i=1Υκ+1(QniKi) ;
(ii) in case µ = 0 , one has
m∑
i=1
‖ap(Dhη)‖p′Lp′(Σ
niKi
)≤ C
h‖Dη‖pLp(Υκ+1(H))
with a constant C independent of ‖η‖E .
Proof of Lemma 8: We consider separately the two cases 1 < p ≤ 2 , p > 2 . Note
the following inequalities, valid for all y1, y2 ∈ (Rd)N (cf. e.g. [DiDT94, Bou97]):
|ap(y1)− ap(y2)|p′ ≤ const |y1 − y2|p, 1 < p ≤ 2;
|ap(y1)− ap(y2)|p′ ≤ const |y1 − y2|p′(|y1|(p−2)p′ + |y2|(p−2)p′
), p ≥ 2.
(70)
a) 1 < p ≤ 2 . We first claim that, i = 1, . . . , m , (70) and (20) yield
‖ap(Dhη)− ap(Dhµ)‖p′Lp′(Σ
niKi
)≤ const
h‖Dh(η − µ)‖p
Lp(QniKi
). (71)
Indeed, note that for arbitrary n and K , for all σ ∈ EK one has
∫ nk
(n−1)k
∫
σ
|Dh(η − µ)|p = d
dK,σ
∫ nk
(n−1)k
∫
S(K,σ)
|Dh(η − µ)|p
by (18); besides, 1/dK,σ ≤ ζ/h by (10).
152 Variational Approach for a Finite Volume Method
Combining (71) with (22), Corollary 1 (cf. the Appendix) and (9), we obtain
‖ap(Dhη)− ap(Dhµ)‖p′Lp′(Σ
niKi
)≤
≤ const
h‖Dh
⊥(η − µ)‖pLp(Υκ(Q
niKi
))≤ const
h‖Dη −Dµ‖p
Lp(Υκ+1(QniKi
)).
Summing over i from 1 to m and using (9) once again, we get (i). Note that the constant
is independent of ‖η‖E, ‖µ‖E , so that (ii) holds in this case.
b) p > 2 . As in case a), we get from (18) and (10)
m∑
i=1
‖ap(Dhη)− ap(Dhµ)‖p′Lp′(Σ
niKi
)≤ const
h
m∑
i=1
‖ap(Dhη)− ap(Dhµ)‖p′Lp′(Q
niKi
).
Further, by (70) and the Holder inequality with q = p/p′ and q′ = (p− 1)/(p− 2) we get
m∑
i=1
‖ap(Dhη)− ap(Dhµ)‖p′Lp′(Σ
niKi
)≤ const
h
( m∑
i=1
‖Dhη −Dhµ‖p′/pLp′(Q
niKi
)
)p′/p×
×( m∑
i=1
‖Dhη‖pLp′(Q
niKi
)+
m∑
i=1
‖Dhµ‖pLp′(Q
niKi
)
)(p−2)/(p−1)
≤
≤ const
hRp(p−2)/(p−1)
( m∑
i=1
‖Dhη −Dhµ‖p′/pLp′(Q
niKi
)
)p′/p.
As in case a), (i) follows by (22), Corollary 1 and (9):
m∑
i=1
‖ap(Dhη)− ap(Dhµ)‖p′Lp′(Σ
niKi
)≤ const
hRp(p−2)/(p−1)‖Dη −Dµ‖p′Lp(Υκ+1(H)).
Moreover, in case µ = 0 we have directly
m∑
i=1
‖ap(Dhη)‖p′Lp′(Σ
niKi
)≤ const
h
m∑
i=1
‖Dhη‖pLp(Q
niKi
)≤ const
h‖Dη‖Lp(Υκ+1(H)),
so that (ii) also holds, which ends the proof. ⋄
Proof of Proposition 3:
(i) Take a grid volume QniKi
⊂ H . We have
Ahη|QniKi
= ap(Dηh)|Qni
Ki
= ap(DηniKi),
where DηniKi
is the solution of the analogue of (35) with AniKi
= ap(Dhη)|∂Ki.
Let ηh be the time average of η in each grid volume:
ηh(t, ·)|QnK=
1
k
∫ nk
(n−1)k
η(τ, ·) dτ for all K, n.
Note that on each QnK , both ηh, ηh do not depend on t . It follows from Lemma 1 that
1
p‖Dηh‖p
Lp(QniKi
)−∫∫
ΣniKi
ηhap(Dhη)νKi≤ 1
p‖Dηh‖p
Lp(QniKi
)−∫∫
ΣniKi
ηhap(Dhη)νKi.
2.II.7. Consistency of the finite volume approximation 153
Summing over i from 1 to m , by the Holder inequality we obtain
‖Dηh‖pLp(H) ≤ ‖Dηh‖pLp(H) + p( m∑
i=1
‖ηh − ηh‖pLp(Σ
niKi
)
)1/p( m∑
i=1
‖ap(Dhη)‖p′Lp′(Σ
niKi
)
)1/p′(72)
in the sense of traces. Applying the Poincare inequality and the imbedding theorem (cf.
Lemma 11 and Lemma 12 in the Appendix, respectively), we obtain
‖ηh − ηh‖pLp(Σ
niKi
)≤ const
( 1
δ(K)δ(K)p‖Dηh −Dηh‖p
Lp(QniKi
)+
+δ(K)p−1‖Dηh −Dηh‖pLp(Q
niKi
)
)≤ const hp−1‖Dηh −Dηh‖p
Lp(QniKi
).
Note that ‖Dηh‖Lp(H) ≤ ‖Dη‖pLp(H) . Therefore
( m∑
i=1
‖ηh − ηh‖pLp(Σ
niKi
)
)1/p≤ const h1/p
′(‖Dηh‖pLp(H) + ‖Dη‖pLp(H)
)1/p. (73)
Taking into account Lemma 8(ii) and applying twice the Young inequality ab ≤ δap+C(δ)bp′,
valid for all a, b ∈ R+, δ > 0 and the corresponding C(δ) , we get from (72),(73)
‖Dηh‖pLp(H) ≤ ‖Dη‖pLp(H)+
+const h1/p′(‖Dηh‖pLp(H) + ‖Dη‖pLp(H)
)1/ph−1/p′‖Dη‖p/p′Lp(Υκ+1(H)) ≤
≤ 1
2‖Dηh‖pLp(H) + const ‖Dη‖pLp(Υκ+1(H));
whence (i) is immediate.
(ii) First note that, by (i), we can assume
‖Dη‖Lp(Q), ‖Dηh‖Lp(Q), ‖Dµ‖Lp(Q), ‖Dµh‖Lp(Q) ≤ R (74)
for some R > 0 , uniformly in h .
Fix a grid volume QnK . Taking (ηh−µh) as a test function in (35) written for ηh , then in
(35) written for µh , subtracting the two identities and integrating in t over ((n−1)k, nk) ,
we get∫∫
QnK
(ap(Dηh)− ap(Dµ
h)) : (Dηh −Dµh) = − 1
km(K)
∫∫
ΣnK
(ap(Dhη)− ap(Dhµ)) :
:
∫∫
QnK
(ηh − µh) +
∫∫
ΣnK
(ηh − µh)(ap(Dhη)− ap(Dhµ))νK .(75)
Consider separately the two cases 1 < p ≤ 2 and p > 2 . Note the following inequalities,
valid for all y1, y2 ∈ (Rd)N (e.g., cf. [DiDT94, Bou97]):
|ap(y1)− ap(y2)|p′ ≤ const (ap(y1)− ap(y2)) : (y1 − y2), 1 < p ≤ 2;
|ap(y1)− ap(y2)|p′ ≤≤ const
[(ap(y1)− ap(y2)) : (y1 − y2)
]p′/2[|y1|p + |y2|p
](2−p′)/2
, p ≥ 2.
(76)
154 Variational Approach for a Finite Volume Method
a) 1 < p ≤ 2 . This time, let us normalize ηh, µh on QnK so that
∫∫
QnK
(ηh − µh) = 0 .
By (75) and (76), we obtain
∫∫
Q
|ap(Dηh)− ap(Dµh)|p′ ≤
≤ const(∑
K,n
‖ap(Dhη)− ap(Dhµ)‖p′Lp′(Σn
K)
)1/p′(∑
K,n
‖ηh − µh‖pLp(ΣnK)
)1/p′.
As in the proof of (i), we find out that
∑
K,n
‖ηh − µh‖pLp(ΣnK) ≤ const h1/p
′‖Dηh −Dµh‖pLp(Q). (77)
On the other hand, Lemma 8(ii) yields
∑
K∈T h
‖ap(Dηh)− ap(Dµh)‖p′Lp(Σn
K) ≤C(R)
h‖Dη −Dµ‖pLp(Q). (78)
Substituting (77),(78) into (75) and taking into account (74), we deduce that
‖ap(Dηh)− ap(Dµh)‖p′
Lp′ (Q)≤ C(R) h−1/p′‖Dη −Dµ‖p/p′Lp(Q)×
×h1/p′‖Dηh −Dµh‖Lp(Q) ≤ C(R)‖Dη −Dµ‖p/p′Lp(Q).
Thus Ah are locally Holder equi-continuous with α = p/p′2 .
b) p > 2 . Using (76),(75),(74), and applying the Holder inequality with q = 2/p′ ,
q′ = 2/(2− p′) , we infer
∫∫
Q
|ap(Dηh)− ap(Dµh)|p′ ≤
(∫∫
Q
(ap(Dηh)− ap(Dµ
h)) : (Dηh −Dµh))p′/2
×
×(|Dηh|p + |Dµh|p
)(2−p′)/2
≤ C(R)(∑
K,n
‖ap(Dhη)− ap(Dhµ)‖p′Lp′(Σn
K)
)1/p′×p′/2
×
×(∑
K,n
‖ηh − µh‖pLp′(Σn
K)
)1/p×p′/2
.
As in case a), we deduce
‖ap(Dηh)− ap(Dµh)‖p′
Lp′(Q)≤ C(R)‖Dη −Dµ‖p′/2Lp(Q).
Thus Ah are locally Holder equi-continuous with α = 1/2 . ⋄
2.II.8. Appendix 155
Appendix: auxiliary results
In this appendix, we give some useful auxiliary results. First, we prove a discrete version of
the Poincare inequality for proportional meshes.
Lemma 9 (Discrete Poincare inequality) Let T be an admissible mesh of Ω ⊂ Rd in
the sense of Definition 2, δ(Ω) = diamΩ , h = size(T ) , and ζ be defined by (10). Let
(wK)K be a set of values in RN such that wK = 0 for all K ∈ Text . Then there exists
α > 0 depending only on p , d and ζ such that
∑
K∈T h
m(K) |wK |p ≤1
αδ(Ω)p
∑
(K,L)∈Υ
1
dm(K|L) dK,L
∣∣∣∣wL − wK
dK,L
∣∣∣∣p
.
Proof : We follow the proof of the Poincare inequality for W 1,p0 (Ω) . Without loss of
generality, one can assume that Ω ⊂ [0, δ(Ω)]d . As in the proof of Lemma 3, for all x ∈ Ω
consider the intersection of the segment [x, x+∆x] with the control volumes of T , where
∆x is chosen to be (−δ(Ω), 0, . . . , 0) ∈ Rd . Let Bx,x+∆x be the corresponding broken line,
and lx,x+∆x its length; one has lx,x+∆x ≤ l(δ(Ω)) in the notation of Lemma 3. Define w
by w|K = wK . Since the boundary condition on w is zero, as in Lemma 3 we have
|w(x)|p ≤ l(δ(Ω))p−1∑
(K,L)∈Υ | [x,x+∆x]∩(K|L)6=Ø
dK,L
∣∣∣∣wK − wL
dK,L
∣∣∣∣p
.
This time, |ΩKL| ≤ δ(Ω)m(K|L) , where ΩKL = x ∈ Rd | [x, x + ∆x] ∩ (K|L) 6= Ø .
Hence∫
Ω
|w|p ≤ l(δ(Ω))p−1∑
(K,L)∈Υ
dK,L
∣∣∣∣wK − wL
dK,L
∣∣∣∣p
|ΩKL| ≤
≤ δ(Ω)l(δ(Ω))p−1d∑
(K,L)∈Υ
1
dm(K|L) dK,L
∣∣∣∣wK − wL
dK,L
∣∣∣∣p
.
Our claim now follows from Lemma 4. ⋄
Remark 7 As in Remark 6, a more careful estimate of |ΩKL| in the proof of Lemma 9 allows
to bypass (10), in case p ≤ 2 . For p = 2 , this yields the result of [YCGH, Remark 7].
Next, we prove another kind of Poincare inequality, under the form convenient for appli-
cation in finite volume schemes (cf. [EGGHH00, Lemma 6.1]).
Lemma 10 Let K be a volume of an admissible mesh of Ω ⊂ Rd in the sense of Definition 2,
and QnK = ((n− 1)k, nk) ×K be the corresponding element of the space-time grid. Then
there exists a constant C that only depends on p , d , and ζ in (11) such that for all
σ ∈ EK , for all w ∈ Lp((n− 1)k, nk;W 1,p(K)) one has
1
dkm(σ) dK,σ
∣∣∣∣wK − wσ
dK,σ
∣∣∣∣p
≤ C ‖Dw‖pLp(QnK) , (79)
156 Variational Approach for a Finite Volume Method
where wK = 1km(K)
∫∫Qn
Kw and wσ = 1
km(σ)
∫ nk
(n−1)k
∫σw , in the sense of traces. Besides,
the same holds with wK replaced by wC = 1|Cn
K |
∫∫CnKw , where Cn
K is defined in Section 1.
Proof : Set h = δ(K) ; one has ν = ζδ(K)/(2√d) . First, note that
1
dkm(σ) dK,σ
∣∣∣∣wK − wσ
dK,σ
∣∣∣∣p
≤ C0(p, d, ζ)m(σ)
hp−1k |wK − wσ|p.
Further, it follows by the Jensen inequality that
k |wK − wσ|p ≤∫ nk
(n−1)k
|wK(t)− wσ(t)|p,
where wK(t) , wσ(t) are the averages of w(t, ·) over K and σ , respectively, for a.a.
t ∈ ((n− 1)k, nk) . Thus it is sufficient to prove that for all w ∈ W 1,p(K) one has
|wK − wσ|p ≤ consthp
m(K)‖Dw‖pLp(K) (80)
(we abusively keep the same notation, dropping the dependence of w on t ).
Furthermore, without loss of generality we can assume that σ is parallel to the hyperplane
x1 = 0 of Rd . Let C = C1 × Cd−1 be a d -dimensional cube with edge 2ν and sides
parallel or perpendicular to σ , contained in K , and such that dist (C, σ) ≥ ν (see fig.2).
Let m(C) denote the d -dimensional measure of C ; set wC = 1m(C)
∫Cw . It is known that
|wK−wC |p ≤ const hp
m(K)‖Dw‖Lp(K) (cf. (84) in Lemma 11 below). Therefore it is sufficient
to prove (80) with wK replaced by wC ; this will also prove the last statement of the lemma.
We denote a point of σ by s and a point of C by x = (ρ, l) , ρ ∈ C1 , l ∈ Cd−1 . By
the standard density argument, we can assume that w ∈ C1(K) . By the Newton-Leibnitz
formula and the Holder inequality one has
|wC − wσ|p ≤1
(m(C)m(σ))p
(∫
C
∫
σ
∫ |x−s|
0
∣∣∣∣Dw(s+ rx− s
|x− s|)∣∣∣∣ drdsdx
)p≤ hp−1
m(C)m(σ)×
×∫
C
∫
σ
∫ |x−s|
0
∣∣∣∣Dw(s+ rx− s
|x− s|)∣∣∣∣p
drdsdx ≤ hp−1
m(σ)
const
hd×
×∫ ∫ ∫
(ρ,l,s,r)∈C1×Cd−1×σ×[0,h]
∣∣∣∣Dw(s+ r
x− s
|x− s|)∣∣∣∣p
drdsdldρ ≤
≤ hp−1
m(σ)
const
hd
(I1 + I2
),
(81)
where I1, I2 are the integrals of |Dw(s+ r x−s|x−s|)|p over M ∩r ∈ [0, ν/2] and M ∩r ∈
[ν/2, h] , respectively, and M = C1 × Cd−1 × σ × [0, h] . In (81) we have extended Dw by
zero outside K , so that all integrals make sense.
Introduce the change of variables (ρ, l, s, r) ↔ (ρ, s, y) , where y = s+ r x−s|x−s| (cf.
fig.2). Clearly, for all (ρ, s) ∈ C1 × σ there is a one-to-one correspondence between (l, r) ∈Cd−1 × [0, h] and y in some subset of K . Moreover, the Jacobian det ‖D(l,r)
Dy‖ of the
2.II.8. Appendix 157
K
σ
C
x ∈ Cρ ∈ C1
l ∈ Cd−1
y ∈ K
s ∈ σr = |y − s|
x1 ∈ R
(x2, . . . , xd) ∈ Rd−1
K
σ
Cyσy
ν/2 ≥ν/2
2ν
Figure 2: The cube C and the change of variables Figure 3: The set σy
transformation (l, r) ↔ y is estimated by const(h/|y1|
)d−1
, where const is independent
of (ρ, l, s, r) ∈M . Thus we have
I2 ≤∫
y∈K
(|Dw(y)|p
∫∫
s∈σ,ρ∈C1
dsdρ
)const dy ≤ hm(σ)‖Dw‖pLp(K). (82)
On the other hand, one has
I1 ≤∫
y∈K
(|Dw(y)|p
∫
s∈σ | ∃x∈C,−→x−s ‖−→y−s
ds
∫
ρ∈C1
dρ
)const
( h
|y1|)d−1
dy
Note that, when |y1| < ν/2 , the (d−1) -dimensional measure of the set σy = s ∈ σ | ∃x ∈C, −−−→x− s ‖−−−→y − s is estimated by const |y1|d−1 , where the constant is absolute (cf. fig.3).
Hence
I2 ≤ const hd−1h
∫
y∈K
|Dw(y)|p 1
|y1|d−1|y1|d−1 dy ≤ const hd ‖Dw‖pLp(K). (83)
Substituting (82),(83) in (81), and taking into account that m(σ) ≤ hd−1 , we finally deduce
(80). This ends the proof. ⋄
Corollary 1 Let (T h, k) be a family of admissible space-time grids in the sense of Defini-
tion 3, and let Dh⊥ be the corresponding operator defined by (19). Let η ∈ E . Then there
exists a constant C which only depends on p , d , and M , ζ such that
∥∥Dh⊥η∥∥pLp(Qn
K)≤ C ‖Dη‖pLp(Υ1(Qn
K)) .
158 Variational Approach for a Finite Volume Method
Proof : Let ηnσ = 1km(σ)
∫ nk
(n−1)k
∫ση . By definition, we have
∥∥Dh⊥η∥∥pLp(Qn
K)=
∑
σ∈EKσ=K|L
1
dkm(σ) dK,σ
∣∣∣∣ηnL − ηnKdK,L
∣∣∣∣p
≤ C∑
σ∈EKσ=K|L
1
dkm(σ) dK,σ
( |ηnσ − ηnK |p(dK,σ)
p +|ηnσ − ηnL|p
dK,σ (dL,σ)p−1
)
≤ C∑
σ∈EK
1
dkm(σ) dK,σ
∣∣∣∣ηnσ − ηnKdK,σ
∣∣∣∣p
+ C∑
σ∈EKσ=K|L
1
dkm(σ) dL,σ
∣∣∣∣ηnσ − ηnLdL,σ
∣∣∣∣p
≤ C ‖Dη‖pLp(Υ1(QnK)) ,
by Lemma 10. ⋄
Finally, we need the following versions of the Poincare inequality the trace imbedding
theorem for the spaces W 1,p .
Lemma 11 Let K ⊂ Rd be convex, bounded of diameter δ(K) and contain a cube C of
edge ζδ(K) > 0 ; let 1 ≤ p < ∞ . Let w ∈ W 1,p(K) , wK = 1|K|
∫Kw . Then there exists
a constant C = C(p, d, ζ) , independent of w , such that
‖w − wK‖Lp(K) ≤ C δ(K) ‖Dw‖Lp(K) .
In addition, one has
|wK − wC|p ≤ Cδ(K)p
|K| ‖Dw‖Lp(K) , (84)
where wC = 1|C|
∫Cw , with a constant C = C(p, d, ζ) , independent of w .
Proof : The proof follows the lines of the proofs of [EgKo, Theorems 59,60], with p = 2
replaced by p ∈ (1,∞) and the Cauchy inequality replaced by general Holder inequality. ⋄
Lemma 12 Let K ⊂ Rd be convex, bounded of diameter δ(K) and contain a ball of radius
ζδ(K) > 0 ; let 1 ≤ p <∞ . Then there exists a constant C which only depends on p and
ζ such that
‖w‖pLp(∂K) ≤ C
(1
δ(K)‖w‖pLp(K) + δ(K)p−1 ‖Dw‖pLp(K)
),
for all w ∈ W 1,p(K) .
2.II.8. Appendix 159
Proof : We cannot refer to [EgKo, Theorem 76], where the boundary of K is supposed
to be Lipschitz. Nevertheless, it is sufficient to introduce coordinates of the spherical type in
a neighbourhood of ∂K of thickness of order δ(K) . More exactly, let xK be the center
of the ball of radius ζδ(K) inside K . Introduce the family of (d−1) -dimensional surfaces
Sτ ⊂ K homothetic to S0 = ∂K with respect to xK . Parametrize it by τ the maximal
distance between the points of Sτ and S0 lying on the ray emanating from xK that have
the longest intersection with K . Note that the surfaces Sτ are well defined at least for
τ ∈ [0, ζδ(K)] . Note also that the distance between the points of Sτ1 and Sτ2 lying on a
same ray emanating from xK does not exceed |τ1− τ2| . The proof of the lemma goes on as
in [EgKo, Theorem 76], upon replacing the original family Sτ by the one constructed above.
⋄
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