Research ArticleBenard Problem for Slightly Compressible Fluids Existence andNonlinear Stability in 3D

Arianna Passerini

Department of Mathematics and Computer Science University of Ferrara Via Machiavelli 30 40121 Ferrara Italy

Correspondence should be addressed to Arianna Passerini ariunifeit

Received 5 May 2020 Accepted 19 June 2020 Published 8 July 2020

Academic Editor Mayer Humi

Copyright copy 2020 Arianna Passerini is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

is paper shows the existence uniqueness and asymptotic behavior in time of regular solutions (a la Ladyzhenskaya) to theBenard problem for a heat-conducting fluid model generalizing the classical OberbeckndashBoussinesq onee novelty of this modelintroduced by Corli and Passerini 2019 and Passerini and Ruggeri 2014 consists in allowing the density of the fluid to alsodepend on the pressure field which as shown by Passerini and Ruggeri 2014 is a necessary request from a thermodynamicviewpoint when dealing with convective problems is property adds to the problem a rather interesting mathematical challengethat is not encountered in the classical model thus requiring a new approach for its resolution

1 Introduction

As is well known one of the most accepted models in thestudy of convection problems in fluid mechanics is the so-called OberbeckndashBoussinesq approximation (hereafterdenoted by OndashB) [1ndash5]e latter is characterized by the factthat one keeps the fluid incompressible and allows fordensity changes only in the buoyancy term of the linearmomentum equation by assuming a linear dependence ontemperature

However as observed in [6] and further elaborated in[7] OndashB may not be entirely satisfactory or even consistentfrom the physical viewpoint [8] if the density is not allowedto have a dependence also on the pressure field p As a resultin [9] a more general theory aimed at removing this in-congruence was proposed More specifically while stillkeeping the incompressibility constraint the density in thebuoyancy term was allowed to depend linearly also on pe remarkable feature of this new model is that now unlikethe classical OndashB the linear momentum equation contains apressure term that is not in the classical gradient form Froma mathematical perspective this feature is very challengingin that some fundamental estimates necessary to ensurestability properties and well-posedness of the relevantproblems [10 11] no longer hold and consequently the

study of such problems requires fresh ideas and differentapproach

With this in mind a systematic study of the mathe-matical properties of the new model was started focusing onthe classical Benard problem As is well known the latterregards convective motions stemming from a motionlessbasic flow r0 of a horizontal layer of fluid heated frombelow For the new model the state r0 is given by [7]

v equiv 0

T(z) equiv Tr(z) T0 minusδT

hz

p(z) equiv pr(z) p0 + pbeminus ρ0gβz

+1β2

αδT

ρ0gh1 minus e

minus ρ0gβz1113872 1113873

minus1β

1 minus eminus ρ0gβz

+αδT

hz1113888 1113889

(1)

with the z coordinate in the vertical direction Here v Tand p are the velocity temperature and pressure fieldsrespectively ρ0 and p0 are the (constant) gauge density andpressure T0 and T0 + δT are the temperature at bottom and

HindawiInternational Journal of Differential EquationsVolume 2020 Article ID 9610689 11 pageshttpsdoiorg10115520209610689

top of the layer and h is the thickness of the layer Finally αand β are the positive material constants Notice that (1)tends to the motionless solution for the same problem in theOndashB approximation by letting β⟶ 0 In [4 9 12] thelinear and nonlinear stability of r0 as well as the existenceand uniqueness for the associated ldquoperturbed problemrdquo wasinvestigated All this work was done in the case of the planeflow

e objective of the current paper is to investigate thesame problem in the more general and appropriate three-dimensional setting e approach followed here to over-come the problem of the pressure mentioned earlier on is tointroduce a new equation of the elliptic type for the pressurefield disregarding the incompressibility constraint (forsteady and for nonlinear equations close to the one of non-Newtonian fluids this approach that allows to achieve aposteriori the divergence-free equation was considered in[5] Actually although this work is done in a ldquoparaboliccontextrdquo (for the kinetic field) the technique in [5] in somesense has inspired the approach in the present paper) Insuch a way the original system of equations is converted tothree highly nonlinearly coupled equations for velocitytemperature and pressure fields However for this newproblem it is shown that if the velocity field v is sufficientlysmooth and also solenoidal at time t 0 then v must besolenoidal in the entire interval of time where the solutionexists thus providing a solution to the original problem Inthis way it is shown that if βlt 2π the relevant perturbedproblem (to the state r0) has a local strong solution (a laLadyzhenskaya) corresponding to initial data of arbitrarysize (in a suitable Sobolev class) Such a solution becomes infact global if this size is appropriately restricted and theRayleigh number is below a certain value see eorem 1

e plan of the paper is given in the following Afterformulating the problem and collecting some preliminaryresults in Section 1 in Section 2 the classical Galerkinmethod is used (see for instance [4]) to construct anldquoapproximaterdquo solution In Section 3 which constitutes themain core of this paper a number of fundamental estimateson the approximate solution is proven that again by classicaltechniques allows us to conclude that the latter converges toa solution of the original problem having the propertiesdescribed above

2 Notations and Preliminary Results

e flow domain is an infinite horizontal layer bounded inthe vertical z-direction with thickness h and unbounded andinvariant in the x and y directions It is assumed that thegeneric perturbed field is periodic in x and y of a givenperiod Lx and Ly respectively an assumption that is justifiedby the fact that the convective roll pattern has a periodicstructure e relevant spatial region thus becomes a ldquope-riodicity cellrdquo defined (in nondimensional form) as

Ω0 (0 a) times(0 b) times(0 1) (2)

with a Lxh and b Lyh Just for simplicity set hereina b 1 e mean value of a function f is indicated as

langfrang 1Ω0

111386811138681113868111386811138681113868111386811138681113946Ω0

f(x y z t)dx dy dz (3)

and a partial derivative with a subscript zξϕ (zϕzξ) ϕξwith ξ isin x y z t1113864 1113865 and similarly for higher-order deriva-tives As customary the inner product in L2(Ω0) is denotedby

(u v) ≔ 1113946Ω0u middot v dx dy dz (4)

where u and v denote the scalar vector or tensor fields ecomponents of a vector u are denoted by u equiv (ux uy uz)Let Wmp(Ω0) be the usual Sobolev space and u(t)mp bethe associated norm e subspace of Wmp(Ω0) constitutedby those u with langurang 0 is denoted by Wmp(Ω0) ByLinfin((0 T) Wmp(Ω0)) we denote the space of functions u

such that

|u|infinmp ≔ ess suptisin[0T]

u(t)mp ltinfin (5)

Likewise for q isin [1infin) the Bochner spaceLq((0 T) Wmp(Ω0)) is the space defined by the norm

|u|qmp ≔ 1113946T

0u(t)

qmpdx1113888 1113889

1q

ltinfin (6)

A decisive role in our stability analysis is played by thePoincare inequality in Ω0

u2 leCnablau2 (7)

As is known (see [13]) set Γ Ω0cap z 0 1 this in-equality is true if at least one of the following cases is sat-isfied (a) functions with a zero mean value (b) vectorfunctions with a vanishing normal component on Γ and (c)scalar functions vanishing on Γ roughout the samesymbol C is used to denote the involved constant whichdepends only on the domain Since boundary conditions areused for which the following integration by parts is valid

nablau2 minus (Δu u) (8)

if the Poincare inequality holds true for u then by theSchwarz inequality it also follows

nablau2 leCΔu2 (9)

As amatter of fact inΩ0 there is an equivalence betweenthe norm of the Laplacian and that of the full set of thesecond derivatives as shown by Ladyzhenskaya (see [14]equations (17) and (18) at page 18])

Δu2 D2u

2 (10)

Further tools herein suitable are some typical 3D im-mersion inequalities the SobolevndashGagliardondashNirenberginequality

u6 leCu12 (11)

which in case the Poincare inequality also holds truebecomes

2 International Journal of Differential Equations

u6 leCnablau2 (12)

nablau6 leCΔu2 (13)

In what follows Morreyrsquos inequality is also used (stillsimplified by the Poincare inequality)

uinfin leCnablau6 (14)

A further preliminary result consists in the estimates forthe Laplace problem (eg [15 16])

Lemma 1 Suppose u is a weak solution to

Δu f inΩ0 with either(a) u 0 or(b) zzu 0 at z 0 1

(15)

en if f isinWm2(Ω0) mge 0 one has u isinWm+22(Ω0) incase (a) and the following inequality holds

um+22 le cfm2 (16)

with c independent of u and f e same result holds in case(b) with u isinWm+22(Ω0) provided f isinWm2(Ω0)

In the present framework once the perturbation fieldsτ T minus Tr and P p minus pr are defined the following non-dimensional system of equations are addressednabla middot v 0

1Pr

zvzt

+ v middot nablav1113888 1113889 minus nablaP minus βPk + Δv + Raτk

zτzt

+ v middot nablaτ Δτ + v middot k

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)

where Ra and Pr are the Rayleigh and Prandtl numbersrespectively and k (0 0 1) is directed along the upwardvertical direction Notice that given p0 the perturbation P

has to be zero at the reference statee boundary conditionsare homogeneous for τ while for v one imposes the im-permeability condition vz 0 and the stress-free conditionvx

z vyz 0 so that one could compare the stability results

with the classic exact ones obtained under the same con-ditions Let us set just for simplicity nabla middot v ϕ

For ldquoregularrdquo solutions ie solutions such thatv isin Linfin

(0 T) W22 Ω0( 11138571113872 1113873capL

2 0 T W32 Ω0( 11138571113872 1113873

vt isin Linfin

(0 T) L2 Ω0( 11138571113872 1113873capL

2 0 T W12 Ω0( 11138571113872 1113873

(18)

corresponding to v0 isinW22(Ω0) at the initial time one canshow that system (17) is equivalent to

ΔP + βPz minus1Prnabla middot (v middot nablav + ϕv) + Raτz

1Pr

zvzt

+ v middot nablav + ϕv1113888 1113889 minus nablaP minus βPk + Δv + Raτk

zτzt

+ v middot nablaτ Δτ + v middot k

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(19)

with Robinrsquos conditions for P at z 0 1

nablaP middot k + βP 0 (20)

e extra nonlinear term lying at the left-hand side of(19)2 is inserted to make the stress-free condition compatiblewith the equation by matching the mean value of thehorizontal components As a matter of fact v middot nablav + ϕv

nabla middot (votimesv) gives rise once integrated to a vanishing boundaryintegral as any other terms in the horizontal projection ofequation (19)2

Actually in one direction the implication is obvious it issufficient to take the divergence of the second equation in(17) to obtain the first of (19)e boundary conditions for P

are established by evaluating at z 0 1 the normal com-ponent of (17)2 the trace of vz

t and by regularity the trace ofvz

zz are defined and vanished leading to (20) As a matter offact by deriving the divergence-free condition with respectto z one gets vx

zx + vyzy + vz

zz 0 and deriving the stress-freecondition along the boundary (since one can exchange theorder of derivation) one gets vx

zx vxxz 0 and

vyzy v

yyz 0 which can be replaced in the previous one

Conversely let us assume that (19) holds true and thesolutions are regular in particular assume thatP isin L2(0 T W22(Ω0)) Let us replace (19)1 in the diver-gence of (19)2 then one obtains

1Pr

zϕzt

Δϕ (21)

Let us remark that langϕrang 0 Indeed if vxx or v

yy is different

from zero then their mean value is zero by periodicity Soone deduces langϕrang langvz

zrang 0 because after integration inz isin (0 1) one obtains vz As a consequence one can use thePoincare inequality for ϕ

Next one can perform the L2-inner product of theequation with ϕ by taking into account the periodicityconditions the right-hand side of (21) becomes

(Δϕϕ) 12

11139461

0dx 1113946

1

0dy ϕ21113872 1113873

z(x y 1) minus ϕ21113872 1113873

z(x y 0)1113960 1113961

minus (nablaϕnablaϕ)

(22)

from the regularity hypotheses one can deduce that theboundary terms vanish in the trace sense because of im-permeability and stress-free conditions which implies vx

zx

vxxz 0 and v

yzy v

yyz 0 On the other hand Robinrsquos

condition for the pressure impliesΔvz 0 in themomentumbalance at the boundary Hence since one can derive theimpermeability condition with respect to x and y it followsalso vz

zz 0 us by summing up one sees that ϕz vanishesat the boundaries

In conclusion one can write

1Pr

ddt

ϕ22 +nablaϕ

22 le 0 (23)

Finally either the divergence of the velocity is zero att 0 and then for all tgt 0 or it decays exponentially to zero

International Journal of Differential Equations 3

is means that the divergence-free condition can be ful-filled and is moreover physically observable

Hereafter a solution with the required regularity isshown to exist provided the initial data for (19)2 and (19)3are in W22(Ω0)

Concerning boundary conditions the change of variableΠ Peβz turns Robinrsquos ones into the simpler Neumannconditions

Πz 0 z 0 1 (24)

and simplifies the system under study which becomes likethe one proposed in [9] page 7 and studied in [4]

ΔΠ minus βΠz minuseβz

Prnabla middot (v middot nablav + ϕv) + Rae

βzτz

1Pr

zvzt

+ v middot nablav + ϕv1113888 1113889 minus Δv minus nabla eminus βzΠ1113872 1113873 minus βe

minus βzΠk

+Raτk

zτzt

+ v middot nablaτ minus Δτ v middot k

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(25)

enew variableΠ is defined up to an arbitrary constantunlike P which appears in the buoyancy force and dependson the boundary value pb which is in the basic solutionus Π can be prescribed with a mean value zero and verifythe Poincare inequality

e trivial solution (v c1i + c2j τ 0 P 0) corre-sponding to the Galilean invariance has to be consideredequivalent to the rest state so for this reason solutions arelooked in Banach subspaces defined by the following condition

langvxrang langvyrang 0 (26)

Next one assigns in Ω0 the periodicity conditions in thehorizontal directions and the stress-free conditions at theboundaries and homogeneous conditions for the deviatorytemperature τ

Πz(x y 0 t) Πz(x y 1 t) 0

vz(x y 0 t) v

z(x y 1 t) 0

vxz(x y 0 t) v

xz(x y 1 t) 0

τ(x y 0 t) τ(x y 1 t) 0

(27)

At the initial time data are given concordant with theprevious ones

(v(x y z 0) τ(x y z 0)) v0(x y z) τ0(x y z)( 1113857

(28)

e plan is studying (17) before proving the equivalencewith (25) To this end one looks for solutions such that aein t and for k isin N0Π belongs to Wk2(Ω0) which is definedas the closure in the Sobolev space norm of Cinfin(Ω0) whosefunctions are periodic in x and y verifies the homogeneousNeumann condition at z 0 1 and has a zero mean value(this last feature is necessary to prove the existence)

Analogously without the isochoric conditionsv equiv (vx vy vz) has free components such that vx and vy

belong toWk2(Ω0) (with the mean value zero as previouslynoticed) while vz and τ belong to Wk2

0 (Ω0) which is theclosure of Cinfin0 (Ω0)

3 Galerkin Approximation

Still in [9] and in [4] an existence result for Π and furtherestimates are given in 2D e proof is unaffected by thedimension and the results are summarized below

Lemma 2 If 0lt βlt 2π and if f isin L2(Ω0) then the problem

ΔΠ minus βΠz eβzf inΩ0

Πz(x y 0) Πz(x y 1) 0 for (x y) isin R2

⎧⎨

⎩ (29)

with periodic side conditions has a unique solutionΠ isinW22(Ω0) such that langΠrang 0 and the following estimateholds true

Π22 leC(β)f2 (30)

In this lemma and hereafter C(β) stands for a positivefunction of β such that limβ⟶0C(β) cge 0 e sameconvention is kept for C(β PrRa)

Furthermore if f nabla middot w and w isin L2(Ω0) with w middot k 0at z 0 1 then in [4] one can also find

Π12 leC(β)w2 (31)

If moreover f nabla middot (nabla middot (uotimesu)) with u middot k 0 at z

0 1 and nabla middot u 0 then in [4] one can also find

Π2 leC(β)u24 (32)

Notice that both these estimates fall into the classicNavierndashStokes estimate

p2 le cv24 (33)

as β goes to zeroHowever here the case nabla middot v ne 0 is in principle

consideredBy taking the data of (29) from system (25) all hy-

potheses concerning the boundary conditions are verifiedand by writing

f minus1Prnabla middot (v middot nablav + ϕv) + Ranabla middot (τk) (34)

one sees that (31) can be usedHence

nablaΠ2 leC(β)1Pr

v middot nablav2 +ϕv2( 1113857 + Raτ21113874 1113875 (35)

Now let us introduce Galerkinrsquos ldquoapproximaterdquo solu-tions of (25) Let us denote by φj1113966 1113967

jisinN and χj1113966 1113967jisinN the

periodic eigenfunctions of the Laplace operator respectivelyverifying Dirichlet and Neumann conditions at zΩ suchthat the vector functions Ψj (χj χjφj) verify the sameboundary condition as v does

4 International Journal of Differential Equations

ΔΨj minus λ(j)Ψj λ(j) gt 0

Δφj minus η(j)φj η(j) gt 0(36)

Moreover Ψjrsquos and φjrsquos are complete orthogonal basesrespectively for the vector and scalar fields in W12(Ω0)Further they belong to W22(Ω0) e Galerkin approxi-mation solutions vN and τN

vN 1113944

N

j1C

jN(t)Ψj

τN 1113944

N

j1A

jN(t)φj

(37)

are then defined in a standard way by projecting system (25)on the N-dimensional subspace spanned by the sameeigenfunctions e ordinary differential equation system soobtained is the first order in the unknowns C

jN(t) and

AjN(t) It is an autonomous homogeneous system in a

normal form whose solutions are easily shown to exist for allN isin N

As a matter of fact although the pressure terms in themomentum balance (which will be written in a compactform as eminus βznablaΠ) cannot be solved a posteriori as forNavierndashStokes equations one can nevertheless define for allN isin N some suitable ΠN as a function of C

j

N(t) and Aj

N(t)To this end let us use (34) in Lemma 2 and denote byA thelinear operator

A ΠN1113872 1113873 ≔ ΔΠN

minus βΠNz e

βznabla

middot minus1Pr

vNmiddot nablavN

+ ϕNvN1113872 1113873 + RaτNk1113874 1113875

(38)

where of course ϕN nabla middot vN By Lemma 2 the operator iscontinuously invertible for 0lt βlt 2π and

ΠN ≔ minus1Pr

1113944

N

ki1A

minus 1eβznabla middot Ψk middot nablaΨi + Ψinabla middot Ψk( 11138571113872 1113873C

kN(t)C

iN(t)

+ RaAminus 11113944

N

i1eβznabla middot φik( 11138571113872 1113873A

iN(t)

(39)

where the summation over repeated indices is understoodAccording to Lemma 2 the linear operatorAminus 1 is defined in

L2(Ω0) and its image belongs to W22(Ω0) Moreover Aminus 1

does not depend on tFor the sake of brevity let us denote

Fki(x y z) ≔ Aminus 1

eβznabla middot Ψk middot nablaΨi + Ψinabla middot Ψk( 11138571113872 1113873

Gi(x y z) ≔ Aminus 1

eβznabla middot φik( 11138571113872 1113873

(40)

Next let us compute

eminus βznablaΠN

Ψj1113872 1113873 minus1Pr

1113944

N

ki1e

minus βznablaFkiΨj1113872 1113873CkNC

iN

+ Ra1113944

N

i1e

minus βznablaGiΨj1113872 1113873AiN

(41)

is expression can be inserted in the Galerkin systemwhich can be written as

_Cj

N minus Prλ(j)CjN minus 1113944

N

ki1Ψk middot nablaΨi minus e

minus βznablaFkiΨj1113872 1113873CkNC

iN+

+ PrRa1113944N

i1φik minus e

minus βznablaGiΨj1113872 1113873AiN

(42)

_Aj

N minus 1113944N

ki1C

kNA

iN Ψk middot nablaφiφj1113872 1113873 minus η(j)A

j

N + Cj

N (43)

It allows for solutions since the right-hand side is aLipschitz continuous function of the variables (in fact it is atmost quadratic) For all N isin N the initial conditionsC

jN(0) C

j0 (v0Ψj) and A

jN(0) A

j0 (τ0Ψj) corre-

spond to the initial conditions of (25) Since the aim is still toderive the equivalence of the differential systems (17) and(19) by showing existence with ϕ 0 the required regularityfor the initial data is W22 In particular one has to define thespace as the closure of linear combinations of the eigen-functions χj minus langχjrang so that vx vy isinW22(Ω0) whilevz τ isinW22

0 (Ω0) which is directly obtained by closing theφjrsquos

Moreover an estimate for vNt (0)2 and τN

t (0)2 ishereafter needed for which the left-hand side of (42)evaluated in t 0 has to be increased by directly inserting(34) in estimate (31)

eminus βznablaΠN

Ψj1113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868le eminus β nablaΠN

Ψj1113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868

leC(β PrRa) vNmiddot nablavN

+ ϕNvNΨj1113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868 + τNΨj1113872 1113873

11138681113868111386811138681113868

111386811138681113868111386811138681113874 1113875

leC(β PrRa) 1113944N

ki1C

kNC

iN Ψk middot nablaΨi + Ψinabla middot ΨkΨj1113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868 + AjN

11138681113868111386811138681113868

11138681113868111386811138681113868⎛⎝ ⎞⎠

(44)

International Journal of Differential Equations 5

After squaring and summing the inequalities derived by(42) and evaluating at the initial time one obtains a boundindependent of N estimated by the W22-norm of the initial

data through Besselrsquos inequality Since the basis is inW22(Ω0) by linearity

ΔvN(0)

2 1113944

N

j1λ(j) v0Ψj1113872 1113873Ψj

2

1113944N

j1v0ΔΨj1113872 1113873Ψj

2

1113944N

j1Δv0Ψj1113872 1113873Ψj

2

le Δv0

2 (45)

Concerning the nonlinear terms one can compute forinstance

1113944

N

j11113944

N

ki11113944

N

ln1Ψk middot nablaΨiΨj1113872 1113873 Ψl middot nablaΨnΨj1113872 1113873C

kN(0)C

iN(0)C

lN(0)C

nN(0)

le 1113944

infin

j1v0 middot nablav0Ψj1113872 1113873

2 v0 middot nablav0

2

(46)

Next these nonlinear terms at the right-hand side can beincreased by (12) and (13) through Holder (p (32) andpprime 3) and interpolation (p (43) and pprime 4) inequalities(the procedure is detailed in the proof of Lemma 3) Finallyusing the Poincare inequality and (10) one gets

vNt (0)

2 leC(β PrRa) Δv0

2 + τ0

2 + Δv0

221113874 1113875

τNt (0)

2 leC(β PrRa) Δτ0

2 + v0

2 + nablav0

2 Δτ0

21113872 1113873

(47)

Here and hereafter the L2-norms of the two nonlinearterms v middot nablav and ϕv are estimated exactly in the same waybut for brevity only the first one will be developed

4 Evolutive Estimates and Main Results

Let us formally derive the a priori estimates for solutions ofsystem (25) by quite a standard procedure they are verifiedjust by Galerkin solutions denoted here by (v τ) and asseen are bound uniformly with respect to N

Lemma 3 Let us define the energy functions

E1(t) ≔12

v22Pr

+ Raτ221113888 1113889

E2(t) ≔12

nablav22Pr

+ Ranablaτ221113888 1113889

(48)

en for sufficiently small εgt 0 the following inequalitieshold true

dE1

dt+ c nablav

22 + Ranablaτ

221113872 1113873leC(β PrRa) E

31 + E

322 + εΔv

221113872 1113873 + C(β Pr)

Raε

v22 (49)

dE2

dt+ c Δv

22 + RaΔτ

221113872 1113873leC(β PrRa)E

32 + C(β Pr)

Raε

E1 (50)

dE2

dt+ c Δv

22 + Raτ

221113872 1113873leC(β PrRa)E

32 + C(β Pr)

Raε

Δv22 (51)

where c isin (0 1) Proof One can test (25) with (v τ) and then one multipliesthe third equation by Ra and then sums

ddt

E1(t) +nablav22 + Ranablaτ

22 minus e

minus βznablaΠ v1113872 1113873 + 2Ra τ vz

( 1113857

minus1Pr

(v middot nablav v) minus1Pr

(ϕv v) minus Ra(v middot nablaτ τ)

(52)

6 International Journal of Differential Equations

In order to estimate the nonlinear term arising from thecoupling with the pressure equation one uses (35) thenonlinear term at the right-hand side can be increased by

(12) and (13) through Holder (p (32) and pprime 3) andinterpolation (p (43) and pprime 4) inequalities so that

eminus βznablaΠ v1113872 1113873

11138681113868111386811138681113868τ0le cv middot nablav2v2 le cv3nablav6v2 le cv

122 v

126 Δv2v2

le cv322 nablav

122 Δv2 le

c

4ε2v

62 +

cε4

nablav22 +

cε2

Δv22

(53)

is term is multiplied by β(C(β)Pr) while the linearpart of the pressure estimate can be included in the estimateof the buoyancy term and of the energy equation transportterm giving rise (in the whole) to

RaC(β Pr)τ2v2 leRaC(β Pr)ε2τ

22 +

12ε

v221113874 1113875 (54)

Here the termwith the second derivatives does not allowto get an autonomous bound for E1(t) so that a furtherestimate for higher derivatives need to be added Notice thatthe convective terms do not vanish and by also resorting toYoungrsquos inequality they are bounded as follows

(v middot nablav v) 1113946Ω0nabla middot v

v2

21113888 1113889 minus 1113946

Ω0ϕ

v2

2le12ϕ2v

24

leCnablav2v122 v

326 leCnablav

32

(v middot nablaτ τ) 1113946Ω0nabla middot v

τ2

21113888 1113889 minus 1113946

Ω0ϕτ2

2le12ϕ2τ

24

leCnablav2τ122 τ

326 leCnablav2nablaτ

22 le

13

Cnablav32 +

23

Cnablaτ32

(55)

Analogously

1113946Ω0ϕv

2 le nablav32 (56)

By collecting all these estimates using the Poincareinequality increasing nablav32 with E32

2 and absorbing at the

left-hand side where possible the terms of order ϵ the firstinequality follows

Now the balance equations are tested with Δv and Δτrespectively and again (35) is inserted

First the following equations are written

12Pr

ddt

nablav22 +Δv

22

1Pr

(v middot nablav + ϕvΔv) + eminus βznablaΠΔv1113872 1113873 minus Ra(τkΔv)

12ddt

nablaτ22 +Δτ

22 (v middot nablaτΔτ) + v

zΔτ( 1113857

dE2

dt+Δv

22 + RaΔτ

22 e

minus βznablaΠΔv1113872 1113873 + Ra τΔvz( 1113857 + Ra Δτ v

z( 1113857

1Pr

(v middot nablavΔv) +1Pr

(ϕvΔv) + Ra(v middot nablaτΔτ)

(57)

Again the pressure-related convective terms arebounded and then the temperature-depending couplingterms are estimated

eminus βznablaΠΔv1113872 1113873

11138681113868111386811138681113868τ0le cv middot nablav2Δv2 le v6nablav3Δv2 le

cv6nablav122 D

2v

122 Δv2 le cnablav

322 Δv

322 le

3cε4

Δv22 +

c

4εnablav

62

(58)

International Journal of Differential Equations 7

In order to reach local-in-time existence of solutions forany initial data one writes

RaC(β Pr) τ2Δv2 +Δτ2v2( 1113857leRaC(β Pr)ε2Δτ

22 +

ε2Δv

22 +

12ε2

E1(t)1113874 1113875 (59)

In order to achieve global-in-time existence for smallinitial data one uses an alternative estimate which followsby several applications of the Poincare inequality

RaC(β Pr) τ2Δv2 +Δτ2v2( 1113857

leRaC(β Pr)ε2Δτ

22 +

12ε

Δv221113874 1113875

(60)

Now the nonlinear terms are focused

(v middot nablav + ϕvΔv)leCv6nablav3Δv2 leCv6nablav122 nablav

126 Δv2

leCnablav322 Δv

322 le

3ε4

CΔv22 +

14ε

Cnablav62

(v middot nablaτΔτ)le v6nablaτ3Δτ2 leCnablav2nablaτ122 D

2τ

122 Δτ2

leCnablav2nablaτ122 Δτ

322 le

3ε4

CΔτ22 +

14ε

Cnablaτ22nablav

42

le3ε4

CΔτ22 +

112ε

Cnablaτ62 +

16ε

Cnablav62

(61)

e terms of order ϵ are still absorbed at the left-handside and then inequalities (50) and (51) respectively followby (59) and (60)

Lemma 4 Let

T ≔ logE2(0) + 1

E2(0)1113888 1113889

14C

(62)

then for all t isin (0 T] the following estimates hold

E(t)le E(0)

1 + E2(0)

1113968

1 + E(0) E(0) minus1 + E2(0)

11139681113872 1113873

⎛⎝ ⎞⎠

12

(63)

where E(t) ≔ E1(t) + E2(t) Furthermore

1113946T

0P(t)dtleC log

E2(0) + 1E2(0)

1113888 1113889

14

E(0) + E12

(0)1113872 1113873

(64)

where P(t) ≔ nablav(t)22 + Δv(t)22 + nablaτ(t)22 + Δτ(t)22

Proof By summing side-by-side the first and second in-equalities in Lemma 3 one obtains

ddt

E(t) + c1P(t)le c2 E(t) + E32

(t) + E3(t)1113872 1113873 (65)

Here one has

E32

E34

E34 le

34

E +14E3 (66)

and consequently the right-hand side can be bounded byE + E3 Hence

1113946E(t)

E(0)

dE

E E2 + 1( ) 1113946

E(t)

E(0)

1E

minusE

E2 + 11113874 1113875dEleCt (67)

then

E(t)E2(0) + 1

1113968

E(0)E2(t) + 1

1113968 le eCt

(68)

It follows that once the maximal interval correspondingto a vanishing denominator is identified

forallt isin (0 T)with 2T ≔ logE2(0) + 1

E2(0)1113888 1113889

12C

(69)

E2(t)le

e2CtE2(0)

1 + E2(0) 1 minus e2Ct( )

leE(0)

1 + E2(0)

1113968

1 + E(0) E(0) minus1 + E2(0)

11139681113872 1113873

(70)

8 International Journal of Differential Equations

the energy is boundedNext coming back to (65) and integrating in [0 T) one

deduces a bound for the norm in Linfin(0 T W12(Ω0))capL2(0 T W22(Ω0)) as can be seen hereafter the inequality

1113946T

0P(t)dtleC log

E2(0) + 1E2(0)

1113888 1113889

14

E(0) + E12

(0)1113872 1113873

(71)

implies in particular the integrability of nablav(t)22 and ofΔv(t)22

Although the Galerkin solutions exist for all tgt 0 theiruniform estimates exist only in the finite interval defined in(69) Global-in-time solution will be found only by askingsuitable smallness of Ra and of the initial data

Finally (25) is derived formally with respect to t and thesame procedure is repeated as in the first estimate but for thenonlinear convective terms Now in order to bound theinitial data for vt initial data in W22 is chosen

ΔΠt minus βΠtz minuseβz

Prnabla middot vt middot nablav + ϕtv + v middot nablavt + ϕvt( 1113857

+Raeβzτtz

1Pr

zvt

zt+ vt middot nablav + ϕtv + v middot nablavt + ϕvt1113888 1113889 minus Δvt

minus nabla eminus βzΠt( 1113857 minus βeminus βzΠtk + Raτtk

zτt

zt+ vt middot nablaτ + v middot nablaτt minus Δτt v

zt

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(72)

Again the terms depending on ϕ or ϕt will be boundedexactly as the corresponding convective ones

Lemma 5 Let us define the energy function as

E(t) ≔12

vt

22

Pr+ Ra τt

22

⎛⎝ ⎞⎠ (73)

en for sufficiently small εgt 0 the following inequalityholds truedEdt

(t) + c nablavt

22 + Ra nablaτt

221113874 1113875leC(β PrRa)Δv

22E(t)

+ C(βPr)Raε

vt

22

(74)

where c isin (0 1)

Proof Let us start with the equation

ddt

E(t) + nablavt

22 + Ra nablaτt

22 minus e

minus βznablaΠt vt1113872 1113873 + 2Ra τt vzt( 1113857 +

minus1Pr

vt middot nablav + v middot nablavt vt( 1113857 minus1Pr

ϕtv + ϕvt vt( 1113857 minus Ra vt middot nablaτ + v middot nablaτt τt( 1113857

(75)

Technically the estimates of the nonlinear convectiveterms are the only different ones from the previous onessince the terms are not homogeneous

eminus βznablaΠt vt1113872 1113873

11138681113868111386811138681113868τt0le c vt

2 vt middot nablav

2 + v middot nablavt

21113872 1113873

le c vt

2 vt

3nablav6 + vt

2vinfin nablavt

21113872 1113873

le c vt

322 nablavt

122 Δv2 + vt

2Δv2 nablavt

21113874 1113875

le c Δv22 vt

22 + ε nablavt

221113874 1113875

(76)

e same estimates can be repeated for the other non-linear terms such as

ϕvt vt( 1113857le cnablav6 vt

2 vt

3

ϕtv vt( 1113857le c vt

2vinfin nablavt

2

vt middot nablaτ τt( 1113857le cnablaτ6 vt

2 τt

3

v middot nablaτt τt( 1113857le c τt

2vinfin nablaτt

2

(77)

e gradient terms can be absorbed at the left-handside

Starting from (74) where the positive term at the left-hand side can be neglected and the term vt

22 at the right-

International Journal of Differential Equations 9

hand side can be bounded by E(t) by a generalizedGronwall argument one shows that

E(t)le ec 1113938

T

0Δv(t)22+1( )dt

E(0) (78)

As a matter of fact (71) implies v isin L2(0 T W22(Ω0))so thatE(t) is bounded in (0 T) Further by coming back to(74) and integrating again it follows also

1113946T

0nablavt

22 + nablaτt

221113874 1113875dtleO v0

22 τ0

221113872 1113873 (79)

and this is independent of the size of Ra

Theorem 1 Let βlt 2π and Pr and Ra be arbitrary thensystem (25) with initial data (v0 τ0) isinW22(Ω0) admits asolution a la Ladyzhenskaya namely

(v τ) isin Linfin 0 T W

22 Ω0( 11138571113872 1113873capL2 0 T W

32 Ω0( 11138571113872 1113873

vt τt( 1113857 isin Linfin 0 T L

2 Ω0( 11138571113872 1113873cap L2 0 T W

12 Ω0( 11138571113872 1113873(80)

where

T ≔ logE2(0) + 1

E2(0)1113888 1113889

14C

(81)

Moreover if nabla middot v0 0 then nabla middot v(t) 0 for all t isin (0 T]Finally if Ra v022 and τ022 are sufficiently smalldepending on Pr and β then T infin is taken in (80) and (v τ)

decays exponentially fast to zero

Proof Employing the estimates proved in Lemma 4 it isroutine to show that the approximate solutions constructedwith the Galerkin method converge to a solution

(v τ) isin Linfin 0 T W

12 Ω0( 11138571113872 1113873capL2 0 T W

22 Ω0( 11138571113872 1113873

Π isin L2 0 T W

22 Ω0( 11138571113872 1113873

vt τt( 1113857 isin Linfin 0 T L

2 Ω0( 11138571113872 1113873cap L2 0 T W

12 Ω0( 11138571113872 1113873

(82)

where T is given in (81)e proof of the further regularity properties implying

that the solution fulfils system (17) too is based on (78) and(79) and in particular on the uniform boundedness of vt2as well as the integrability of nablavt(t)22 To this end viewingthe momentum balance as a Poisson problem and increasingthe pressure in the usual way the following is obtained withthe help of Lemma 1

v(t)22 leC v middot nablav2 +nablaΠ2 + vt

2 +τ21113872 1113873

leC nablav42 + εΔv

22 + vt

2 +τ21113872 1113873

(83)

e ε term at the right-hand side can be absorbed by theleft-hand side so that finally the following bound is obtained

ess sup(0T)

v(t)22 leCess sup(0T)

nablav(t)42 +τ2 + vt

21113872 1113873

leO v0

22 τ0

221113872 1113873

(84)

Moreover again from Lemma 1 it is also obtained that

1113946T

0v(t)

232dtleC 1113946

T

0v middot nablav

212 +Π

222 + vt

212 +τ

2121113874 1113875dt

(85)

It is remarked that as noticed in Section 2 the com-patibility condition related to the Neumann problem isverified Here the nonlinear terms at the right-hand side canbe increased by using (14) and interpolation inequality

v middot D2v

2 le cvinfin D

2v

2 le c D2v

22

nablav(nablav)T

le cnablav

24 le cnablav6nablav3 le c D

2v

22

(86)

e proof that τ is bounded in Linfin(0 T W12(Ω0))capL2(0 T W22(Ω0)) is similar and simpler

e proof of global existence follows from Lemmas 3 and5 by setting

E0(t) ≔ E(t) + E(t)

P0(t) ≔ nablav(t)22 +Δv(t)

22 + nablavt(t)

22

+ Ra nablaτ(t)22 +Δτ(t)

22 + nablaτt(t)

221113874 1113875

(87)

In fact if inequalities (49) (51) and (74) are addedbringing to the left-hand side the three terms with factorRaε and using the Poincare inequality

Ele kP (88)

to absorb them in P0 for sufficiently small Ra (dependingon ε) one gets

ddt

E(t) + c1P(t)le c2 E32

(t) + E3(t)1113872 1113873 (89)

e constant c1 is now fixed positive by suitablyrestricting Ra Again using the Poincare inequality on theleft-hand side of (89) one infers

ddt

E(t) + c1 minus c2 E12

(t) + E2(t)1113872 11138731113960 1113961

1k

E(t)le 0 (90)

us the exponential decay follows by choosing at theinitial time

E12

(0) + E2(0)le

12

c1

c2 (91)

In fact by continuity for some time E(t)leE(0) Butthis reinforces the previous condition so that the positivefunction in square brackets cannot decrease and is boundedfrom below by a positive constant for all times

Data Availability

e data that support the findings of this study are availablewithin the article

Conflicts of Interest

e author declares that there are no conflicts of interest

10 International Journal of Differential Equations

References

[1] A Giacobbe and G Mulone ldquoStability in the rotating Benardproblem and its optimal Lyapunov functionsrdquo Acta Appli-candae Mathematicae vol 132 no 1 2014

[2] A Passerini C Ferrario M Ruzicka and G ater ldquoe-oretical results on steady convective flows between horizontalcoaxial cylindersrdquo SIAM Journal on Applied Mathematicsvol 71 no 2 pp 465ndash486 2011

[3] A Passerini and C Ferrario ldquoA theoretical study of the firsttransition for the non-linear Stokes problem in a horizontalannulusrdquo International Journal of Non-linear Mechanicsvol 78 no 1 pp 1ndash8 2015

[4] A De Martino and A Passerini ldquoExistence and nonlinearstability of convective solutions for almost compressible fluidsin Benard problemrdquo Journal of Mathematical Physics vol 60no 11 Article ID 113101 2019

[5] F Crispo and P Maremonti ldquoA high regularity result ofsolutions to modified p-Stokes equationsrdquo Nonlinear Anal-ysis vol 118 pp 97ndash129 2015

[6] H Gouin and T Ruggeri ldquoA consistent thermodynamicalmodel of incompressible media as limit case of quasi-thermal-incompressible materialsrdquo International Journal of Non-Linear Mechanics vol 47 no 6 pp 688ndash693 2012

[7] A Passerini and T Ruggeri ldquoe Benard problem for quasi-thermal-incompressible materials a linear analysisrdquo Inter-national Journal of Non-Linear Mechanics vol 67 no 1pp 178ndash185 2014

[8] S Gatti and E Vuk ldquoSingular limit of equations for linearviscoelastic fluids with periodic boundary conditionsrdquo In-ternational Journal of Non-Linear Mechanics vol 41 no 4pp 518ndash526 2006

[9] A Corli and A Passerini ldquoe Benard problem for slightlycompressible materials existence and linear instabilityrdquoMediterranean Journal of Mathematics vol 16 no 1 2019

[10] G P Galdi ldquoNon-linear stability of the magnetic Benardproblem via a generalized energy methodrdquo Archive for Ra-tional Mechanics and Analysis vol 87 no 2 pp 167ndash1861985

[11] G P Galdi and M Padula ldquoA new approach to energy theoryin the stability of fluid motionrdquo Archive for Rational Me-chanics and Analysis vol 110 no 3 pp 187ndash286 1990

[12] A De Martino and A Passerini ldquoA Lorenz model for almostcompressible fluidsrdquo Mediterranean Journal of Mathematicsvol 17 no 1 p 12 2020

[13] G P Galdi An Introduction to the Mathematical eory of theNavier-Stokes Equation Springer New York NY USA 1998

[14] O Ladyzhenskaya e Mathematical eory of Viscous In-compressible Fluids Gordon amp Breach London UK 1963

[15] G Iooss ldquoeorie non lineaire de la stabilite des ecoulementslaminaires dans le cas de ldquolrsquoechange des stabilitesrdquo Archive forRational Mechanics and Analysis vol 40 no 3 pp 166ndash2081971

[16] M Schechter ldquoCoerciveness in Lprdquo Transactions of theAmerican Mathematical Society vol 107 no 1 pp 10ndash291963

International Journal of Differential Equations 11

u6 leCnablau2 (12)

nablau6 leCΔu2 (13)

In what follows Morreyrsquos inequality is also used (stillsimplified by the Poincare inequality)

uinfin leCnablau6 (14)

A further preliminary result consists in the estimates forthe Laplace problem (eg [15 16])

Lemma 1 Suppose u is a weak solution to

Δu f inΩ0 with either(a) u 0 or(b) zzu 0 at z 0 1

(15)

en if f isinWm2(Ω0) mge 0 one has u isinWm+22(Ω0) incase (a) and the following inequality holds

um+22 le cfm2 (16)

with c independent of u and f e same result holds in case(b) with u isinWm+22(Ω0) provided f isinWm2(Ω0)

In the present framework once the perturbation fieldsτ T minus Tr and P p minus pr are defined the following non-dimensional system of equations are addressednabla middot v 0

1Pr

zvzt

+ v middot nablav1113888 1113889 minus nablaP minus βPk + Δv + Raτk

zτzt

+ v middot nablaτ Δτ + v middot k

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)

where Ra and Pr are the Rayleigh and Prandtl numbersrespectively and k (0 0 1) is directed along the upwardvertical direction Notice that given p0 the perturbation P

has to be zero at the reference statee boundary conditionsare homogeneous for τ while for v one imposes the im-permeability condition vz 0 and the stress-free conditionvx

z vyz 0 so that one could compare the stability results

with the classic exact ones obtained under the same con-ditions Let us set just for simplicity nabla middot v ϕ

For ldquoregularrdquo solutions ie solutions such thatv isin Linfin

(0 T) W22 Ω0( 11138571113872 1113873capL

2 0 T W32 Ω0( 11138571113872 1113873

vt isin Linfin

(0 T) L2 Ω0( 11138571113872 1113873capL

2 0 T W12 Ω0( 11138571113872 1113873

(18)

corresponding to v0 isinW22(Ω0) at the initial time one canshow that system (17) is equivalent to

ΔP + βPz minus1Prnabla middot (v middot nablav + ϕv) + Raτz

1Pr

zvzt

+ v middot nablav + ϕv1113888 1113889 minus nablaP minus βPk + Δv + Raτk

zτzt

+ v middot nablaτ Δτ + v middot k

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(19)

with Robinrsquos conditions for P at z 0 1

nablaP middot k + βP 0 (20)

e extra nonlinear term lying at the left-hand side of(19)2 is inserted to make the stress-free condition compatiblewith the equation by matching the mean value of thehorizontal components As a matter of fact v middot nablav + ϕv

nabla middot (votimesv) gives rise once integrated to a vanishing boundaryintegral as any other terms in the horizontal projection ofequation (19)2

Actually in one direction the implication is obvious it issufficient to take the divergence of the second equation in(17) to obtain the first of (19)e boundary conditions for P

are established by evaluating at z 0 1 the normal com-ponent of (17)2 the trace of vz

t and by regularity the trace ofvz

zz are defined and vanished leading to (20) As a matter offact by deriving the divergence-free condition with respectto z one gets vx

zx + vyzy + vz

zz 0 and deriving the stress-freecondition along the boundary (since one can exchange theorder of derivation) one gets vx

zx vxxz 0 and

vyzy v

yyz 0 which can be replaced in the previous one

Conversely let us assume that (19) holds true and thesolutions are regular in particular assume thatP isin L2(0 T W22(Ω0)) Let us replace (19)1 in the diver-gence of (19)2 then one obtains

1Pr

zϕzt

Δϕ (21)

Let us remark that langϕrang 0 Indeed if vxx or v

yy is different

from zero then their mean value is zero by periodicity Soone deduces langϕrang langvz

zrang 0 because after integration inz isin (0 1) one obtains vz As a consequence one can use thePoincare inequality for ϕ

Next one can perform the L2-inner product of theequation with ϕ by taking into account the periodicityconditions the right-hand side of (21) becomes

(Δϕϕ) 12

11139461

0dx 1113946

1

0dy ϕ21113872 1113873

z(x y 1) minus ϕ21113872 1113873

z(x y 0)1113960 1113961

minus (nablaϕnablaϕ)

(22)

from the regularity hypotheses one can deduce that theboundary terms vanish in the trace sense because of im-permeability and stress-free conditions which implies vx

zx

vxxz 0 and v

yzy v

yyz 0 On the other hand Robinrsquos

condition for the pressure impliesΔvz 0 in themomentumbalance at the boundary Hence since one can derive theimpermeability condition with respect to x and y it followsalso vz

zz 0 us by summing up one sees that ϕz vanishesat the boundaries

In conclusion one can write

1Pr

ddt

ϕ22 +nablaϕ

22 le 0 (23)

Finally either the divergence of the velocity is zero att 0 and then for all tgt 0 or it decays exponentially to zero

International Journal of Differential Equations 3

is means that the divergence-free condition can be ful-filled and is moreover physically observable

Hereafter a solution with the required regularity isshown to exist provided the initial data for (19)2 and (19)3are in W22(Ω0)

Concerning boundary conditions the change of variableΠ Peβz turns Robinrsquos ones into the simpler Neumannconditions

Πz 0 z 0 1 (24)

and simplifies the system under study which becomes likethe one proposed in [9] page 7 and studied in [4]

ΔΠ minus βΠz minuseβz

Prnabla middot (v middot nablav + ϕv) + Rae

βzτz

1Pr

zvzt

+ v middot nablav + ϕv1113888 1113889 minus Δv minus nabla eminus βzΠ1113872 1113873 minus βe

minus βzΠk

+Raτk

zτzt

+ v middot nablaτ minus Δτ v middot k

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(25)

enew variableΠ is defined up to an arbitrary constantunlike P which appears in the buoyancy force and dependson the boundary value pb which is in the basic solutionus Π can be prescribed with a mean value zero and verifythe Poincare inequality

e trivial solution (v c1i + c2j τ 0 P 0) corre-sponding to the Galilean invariance has to be consideredequivalent to the rest state so for this reason solutions arelooked in Banach subspaces defined by the following condition

langvxrang langvyrang 0 (26)

Next one assigns in Ω0 the periodicity conditions in thehorizontal directions and the stress-free conditions at theboundaries and homogeneous conditions for the deviatorytemperature τ

Πz(x y 0 t) Πz(x y 1 t) 0

vz(x y 0 t) v

z(x y 1 t) 0

vxz(x y 0 t) v

xz(x y 1 t) 0

τ(x y 0 t) τ(x y 1 t) 0

(27)

At the initial time data are given concordant with theprevious ones

(v(x y z 0) τ(x y z 0)) v0(x y z) τ0(x y z)( 1113857

(28)

e plan is studying (17) before proving the equivalencewith (25) To this end one looks for solutions such that aein t and for k isin N0Π belongs to Wk2(Ω0) which is definedas the closure in the Sobolev space norm of Cinfin(Ω0) whosefunctions are periodic in x and y verifies the homogeneousNeumann condition at z 0 1 and has a zero mean value(this last feature is necessary to prove the existence)

Analogously without the isochoric conditionsv equiv (vx vy vz) has free components such that vx and vy

belong toWk2(Ω0) (with the mean value zero as previouslynoticed) while vz and τ belong to Wk2

0 (Ω0) which is theclosure of Cinfin0 (Ω0)

3 Galerkin Approximation

Still in [9] and in [4] an existence result for Π and furtherestimates are given in 2D e proof is unaffected by thedimension and the results are summarized below

Lemma 2 If 0lt βlt 2π and if f isin L2(Ω0) then the problem

ΔΠ minus βΠz eβzf inΩ0

Πz(x y 0) Πz(x y 1) 0 for (x y) isin R2

⎧⎨

⎩ (29)

with periodic side conditions has a unique solutionΠ isinW22(Ω0) such that langΠrang 0 and the following estimateholds true

Π22 leC(β)f2 (30)

In this lemma and hereafter C(β) stands for a positivefunction of β such that limβ⟶0C(β) cge 0 e sameconvention is kept for C(β PrRa)

Furthermore if f nabla middot w and w isin L2(Ω0) with w middot k 0at z 0 1 then in [4] one can also find

Π12 leC(β)w2 (31)

If moreover f nabla middot (nabla middot (uotimesu)) with u middot k 0 at z

0 1 and nabla middot u 0 then in [4] one can also find

Π2 leC(β)u24 (32)

Notice that both these estimates fall into the classicNavierndashStokes estimate

p2 le cv24 (33)

as β goes to zeroHowever here the case nabla middot v ne 0 is in principle

consideredBy taking the data of (29) from system (25) all hy-

potheses concerning the boundary conditions are verifiedand by writing

f minus1Prnabla middot (v middot nablav + ϕv) + Ranabla middot (τk) (34)

one sees that (31) can be usedHence

nablaΠ2 leC(β)1Pr

v middot nablav2 +ϕv2( 1113857 + Raτ21113874 1113875 (35)

Now let us introduce Galerkinrsquos ldquoapproximaterdquo solu-tions of (25) Let us denote by φj1113966 1113967

jisinN and χj1113966 1113967jisinN the

periodic eigenfunctions of the Laplace operator respectivelyverifying Dirichlet and Neumann conditions at zΩ suchthat the vector functions Ψj (χj χjφj) verify the sameboundary condition as v does

4 International Journal of Differential Equations

ΔΨj minus λ(j)Ψj λ(j) gt 0

Δφj minus η(j)φj η(j) gt 0(36)

Moreover Ψjrsquos and φjrsquos are complete orthogonal basesrespectively for the vector and scalar fields in W12(Ω0)Further they belong to W22(Ω0) e Galerkin approxi-mation solutions vN and τN

vN 1113944

N

j1C

jN(t)Ψj

τN 1113944

N

j1A

jN(t)φj

(37)

are then defined in a standard way by projecting system (25)on the N-dimensional subspace spanned by the sameeigenfunctions e ordinary differential equation system soobtained is the first order in the unknowns C

jN(t) and

AjN(t) It is an autonomous homogeneous system in a

normal form whose solutions are easily shown to exist for allN isin N

As a matter of fact although the pressure terms in themomentum balance (which will be written in a compactform as eminus βznablaΠ) cannot be solved a posteriori as forNavierndashStokes equations one can nevertheless define for allN isin N some suitable ΠN as a function of C

j

N(t) and Aj

N(t)To this end let us use (34) in Lemma 2 and denote byA thelinear operator

A ΠN1113872 1113873 ≔ ΔΠN

minus βΠNz e

βznabla

middot minus1Pr

vNmiddot nablavN

+ ϕNvN1113872 1113873 + RaτNk1113874 1113875

(38)

where of course ϕN nabla middot vN By Lemma 2 the operator iscontinuously invertible for 0lt βlt 2π and

ΠN ≔ minus1Pr

1113944

N

ki1A

minus 1eβznabla middot Ψk middot nablaΨi + Ψinabla middot Ψk( 11138571113872 1113873C

kN(t)C

iN(t)

+ RaAminus 11113944

N

i1eβznabla middot φik( 11138571113872 1113873A

iN(t)

(39)

where the summation over repeated indices is understoodAccording to Lemma 2 the linear operatorAminus 1 is defined in

L2(Ω0) and its image belongs to W22(Ω0) Moreover Aminus 1

does not depend on tFor the sake of brevity let us denote

Fki(x y z) ≔ Aminus 1

eβznabla middot Ψk middot nablaΨi + Ψinabla middot Ψk( 11138571113872 1113873

Gi(x y z) ≔ Aminus 1

eβznabla middot φik( 11138571113872 1113873

(40)

Next let us compute

eminus βznablaΠN

Ψj1113872 1113873 minus1Pr

1113944

N

ki1e

minus βznablaFkiΨj1113872 1113873CkNC

iN

+ Ra1113944

N

i1e

minus βznablaGiΨj1113872 1113873AiN

(41)

is expression can be inserted in the Galerkin systemwhich can be written as

_Cj

N minus Prλ(j)CjN minus 1113944

N

ki1Ψk middot nablaΨi minus e

minus βznablaFkiΨj1113872 1113873CkNC

iN+

+ PrRa1113944N

i1φik minus e

minus βznablaGiΨj1113872 1113873AiN

(42)

_Aj

N minus 1113944N

ki1C

kNA

iN Ψk middot nablaφiφj1113872 1113873 minus η(j)A

j

N + Cj

N (43)

It allows for solutions since the right-hand side is aLipschitz continuous function of the variables (in fact it is atmost quadratic) For all N isin N the initial conditionsC

jN(0) C

j0 (v0Ψj) and A

jN(0) A

j0 (τ0Ψj) corre-

spond to the initial conditions of (25) Since the aim is still toderive the equivalence of the differential systems (17) and(19) by showing existence with ϕ 0 the required regularityfor the initial data is W22 In particular one has to define thespace as the closure of linear combinations of the eigen-functions χj minus langχjrang so that vx vy isinW22(Ω0) whilevz τ isinW22

0 (Ω0) which is directly obtained by closing theφjrsquos

Moreover an estimate for vNt (0)2 and τN

t (0)2 ishereafter needed for which the left-hand side of (42)evaluated in t 0 has to be increased by directly inserting(34) in estimate (31)

eminus βznablaΠN

Ψj1113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868le eminus β nablaΠN

Ψj1113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868

leC(β PrRa) vNmiddot nablavN

+ ϕNvNΨj1113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868 + τNΨj1113872 1113873

11138681113868111386811138681113868

111386811138681113868111386811138681113874 1113875

leC(β PrRa) 1113944N

ki1C

kNC

iN Ψk middot nablaΨi + Ψinabla middot ΨkΨj1113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868 + AjN

11138681113868111386811138681113868

11138681113868111386811138681113868⎛⎝ ⎞⎠

(44)

International Journal of Differential Equations 5

After squaring and summing the inequalities derived by(42) and evaluating at the initial time one obtains a boundindependent of N estimated by the W22-norm of the initial

data through Besselrsquos inequality Since the basis is inW22(Ω0) by linearity

ΔvN(0)

2 1113944

N

j1λ(j) v0Ψj1113872 1113873Ψj

2

1113944N

j1v0ΔΨj1113872 1113873Ψj

2

1113944N

j1Δv0Ψj1113872 1113873Ψj

2

le Δv0

2 (45)

Concerning the nonlinear terms one can compute forinstance

1113944

N

j11113944

N

ki11113944

N

ln1Ψk middot nablaΨiΨj1113872 1113873 Ψl middot nablaΨnΨj1113872 1113873C

kN(0)C

iN(0)C

lN(0)C

nN(0)

le 1113944

infin

j1v0 middot nablav0Ψj1113872 1113873

2 v0 middot nablav0

2

(46)

Next these nonlinear terms at the right-hand side can beincreased by (12) and (13) through Holder (p (32) andpprime 3) and interpolation (p (43) and pprime 4) inequalities(the procedure is detailed in the proof of Lemma 3) Finallyusing the Poincare inequality and (10) one gets

vNt (0)

2 leC(β PrRa) Δv0

2 + τ0

2 + Δv0

221113874 1113875

τNt (0)

2 leC(β PrRa) Δτ0

2 + v0

2 + nablav0

2 Δτ0

21113872 1113873

(47)

Here and hereafter the L2-norms of the two nonlinearterms v middot nablav and ϕv are estimated exactly in the same waybut for brevity only the first one will be developed

4 Evolutive Estimates and Main Results

Let us formally derive the a priori estimates for solutions ofsystem (25) by quite a standard procedure they are verifiedjust by Galerkin solutions denoted here by (v τ) and asseen are bound uniformly with respect to N

Lemma 3 Let us define the energy functions

E1(t) ≔12

v22Pr

+ Raτ221113888 1113889

E2(t) ≔12

nablav22Pr

+ Ranablaτ221113888 1113889

(48)

en for sufficiently small εgt 0 the following inequalitieshold true

dE1

dt+ c nablav

22 + Ranablaτ

221113872 1113873leC(β PrRa) E

31 + E

322 + εΔv

221113872 1113873 + C(β Pr)

Raε

v22 (49)

dE2

dt+ c Δv

22 + RaΔτ

221113872 1113873leC(β PrRa)E

32 + C(β Pr)

Raε

E1 (50)

dE2

dt+ c Δv

22 + Raτ

221113872 1113873leC(β PrRa)E

32 + C(β Pr)

Raε

Δv22 (51)

where c isin (0 1) Proof One can test (25) with (v τ) and then one multipliesthe third equation by Ra and then sums

ddt

E1(t) +nablav22 + Ranablaτ

22 minus e

minus βznablaΠ v1113872 1113873 + 2Ra τ vz

( 1113857

minus1Pr

(v middot nablav v) minus1Pr

(ϕv v) minus Ra(v middot nablaτ τ)

(52)

6 International Journal of Differential Equations

In order to estimate the nonlinear term arising from thecoupling with the pressure equation one uses (35) thenonlinear term at the right-hand side can be increased by

(12) and (13) through Holder (p (32) and pprime 3) andinterpolation (p (43) and pprime 4) inequalities so that

eminus βznablaΠ v1113872 1113873

11138681113868111386811138681113868τ0le cv middot nablav2v2 le cv3nablav6v2 le cv

122 v

126 Δv2v2

le cv322 nablav

122 Δv2 le

c

4ε2v

62 +

cε4

nablav22 +

cε2

Δv22

(53)

is term is multiplied by β(C(β)Pr) while the linearpart of the pressure estimate can be included in the estimateof the buoyancy term and of the energy equation transportterm giving rise (in the whole) to

RaC(β Pr)τ2v2 leRaC(β Pr)ε2τ

22 +

12ε

v221113874 1113875 (54)

Here the termwith the second derivatives does not allowto get an autonomous bound for E1(t) so that a furtherestimate for higher derivatives need to be added Notice thatthe convective terms do not vanish and by also resorting toYoungrsquos inequality they are bounded as follows

(v middot nablav v) 1113946Ω0nabla middot v

v2

21113888 1113889 minus 1113946

Ω0ϕ

v2

2le12ϕ2v

24

leCnablav2v122 v

326 leCnablav

32

(v middot nablaτ τ) 1113946Ω0nabla middot v

τ2

21113888 1113889 minus 1113946

Ω0ϕτ2

2le12ϕ2τ

24

leCnablav2τ122 τ

326 leCnablav2nablaτ

22 le

13

Cnablav32 +

23

Cnablaτ32

(55)

Analogously

1113946Ω0ϕv

2 le nablav32 (56)

By collecting all these estimates using the Poincareinequality increasing nablav32 with E32

2 and absorbing at the

left-hand side where possible the terms of order ϵ the firstinequality follows

Now the balance equations are tested with Δv and Δτrespectively and again (35) is inserted

First the following equations are written

12Pr

ddt

nablav22 +Δv

22

1Pr

(v middot nablav + ϕvΔv) + eminus βznablaΠΔv1113872 1113873 minus Ra(τkΔv)

12ddt

nablaτ22 +Δτ

22 (v middot nablaτΔτ) + v

zΔτ( 1113857

dE2

dt+Δv

22 + RaΔτ

22 e

minus βznablaΠΔv1113872 1113873 + Ra τΔvz( 1113857 + Ra Δτ v

z( 1113857

1Pr

(v middot nablavΔv) +1Pr

(ϕvΔv) + Ra(v middot nablaτΔτ)

(57)

Again the pressure-related convective terms arebounded and then the temperature-depending couplingterms are estimated

eminus βznablaΠΔv1113872 1113873

11138681113868111386811138681113868τ0le cv middot nablav2Δv2 le v6nablav3Δv2 le

cv6nablav122 D

2v

122 Δv2 le cnablav

322 Δv

322 le

3cε4

Δv22 +

c

4εnablav

62

(58)

International Journal of Differential Equations 7

In order to reach local-in-time existence of solutions forany initial data one writes

RaC(β Pr) τ2Δv2 +Δτ2v2( 1113857leRaC(β Pr)ε2Δτ

22 +

ε2Δv

22 +

12ε2

E1(t)1113874 1113875 (59)

In order to achieve global-in-time existence for smallinitial data one uses an alternative estimate which followsby several applications of the Poincare inequality

RaC(β Pr) τ2Δv2 +Δτ2v2( 1113857

leRaC(β Pr)ε2Δτ

22 +

12ε

Δv221113874 1113875

(60)

Now the nonlinear terms are focused

(v middot nablav + ϕvΔv)leCv6nablav3Δv2 leCv6nablav122 nablav

126 Δv2

leCnablav322 Δv

322 le

3ε4

CΔv22 +

14ε

Cnablav62

(v middot nablaτΔτ)le v6nablaτ3Δτ2 leCnablav2nablaτ122 D

2τ

122 Δτ2

leCnablav2nablaτ122 Δτ

322 le

3ε4

CΔτ22 +

14ε

Cnablaτ22nablav

42

le3ε4

CΔτ22 +

112ε

Cnablaτ62 +

16ε

Cnablav62

(61)

e terms of order ϵ are still absorbed at the left-handside and then inequalities (50) and (51) respectively followby (59) and (60)

Lemma 4 Let

T ≔ logE2(0) + 1

E2(0)1113888 1113889

14C

(62)

then for all t isin (0 T] the following estimates hold

E(t)le E(0)

1 + E2(0)

1113968

1 + E(0) E(0) minus1 + E2(0)

11139681113872 1113873

⎛⎝ ⎞⎠

12

(63)

where E(t) ≔ E1(t) + E2(t) Furthermore

1113946T

0P(t)dtleC log

E2(0) + 1E2(0)

1113888 1113889

14

E(0) + E12

(0)1113872 1113873

(64)

where P(t) ≔ nablav(t)22 + Δv(t)22 + nablaτ(t)22 + Δτ(t)22

Proof By summing side-by-side the first and second in-equalities in Lemma 3 one obtains

ddt

E(t) + c1P(t)le c2 E(t) + E32

(t) + E3(t)1113872 1113873 (65)

Here one has

E32

E34

E34 le

34

E +14E3 (66)

and consequently the right-hand side can be bounded byE + E3 Hence

1113946E(t)

E(0)

dE

E E2 + 1( ) 1113946

E(t)

E(0)

1E

minusE

E2 + 11113874 1113875dEleCt (67)

then

E(t)E2(0) + 1

1113968

E(0)E2(t) + 1

1113968 le eCt

(68)

It follows that once the maximal interval correspondingto a vanishing denominator is identified

forallt isin (0 T)with 2T ≔ logE2(0) + 1

E2(0)1113888 1113889

12C

(69)

E2(t)le

e2CtE2(0)

1 + E2(0) 1 minus e2Ct( )

leE(0)

1 + E2(0)

1113968

1 + E(0) E(0) minus1 + E2(0)

11139681113872 1113873

(70)

8 International Journal of Differential Equations

the energy is boundedNext coming back to (65) and integrating in [0 T) one

deduces a bound for the norm in Linfin(0 T W12(Ω0))capL2(0 T W22(Ω0)) as can be seen hereafter the inequality

1113946T

0P(t)dtleC log

E2(0) + 1E2(0)

1113888 1113889

14

E(0) + E12

(0)1113872 1113873

(71)

implies in particular the integrability of nablav(t)22 and ofΔv(t)22

Although the Galerkin solutions exist for all tgt 0 theiruniform estimates exist only in the finite interval defined in(69) Global-in-time solution will be found only by askingsuitable smallness of Ra and of the initial data

Finally (25) is derived formally with respect to t and thesame procedure is repeated as in the first estimate but for thenonlinear convective terms Now in order to bound theinitial data for vt initial data in W22 is chosen

ΔΠt minus βΠtz minuseβz

Prnabla middot vt middot nablav + ϕtv + v middot nablavt + ϕvt( 1113857

+Raeβzτtz

1Pr

zvt

zt+ vt middot nablav + ϕtv + v middot nablavt + ϕvt1113888 1113889 minus Δvt

minus nabla eminus βzΠt( 1113857 minus βeminus βzΠtk + Raτtk

zτt

zt+ vt middot nablaτ + v middot nablaτt minus Δτt v

zt

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(72)

Again the terms depending on ϕ or ϕt will be boundedexactly as the corresponding convective ones

Lemma 5 Let us define the energy function as

E(t) ≔12

vt

22

Pr+ Ra τt

22

⎛⎝ ⎞⎠ (73)

en for sufficiently small εgt 0 the following inequalityholds truedEdt

(t) + c nablavt

22 + Ra nablaτt

221113874 1113875leC(β PrRa)Δv

22E(t)

+ C(βPr)Raε

vt

22

(74)

where c isin (0 1)

Proof Let us start with the equation

ddt

E(t) + nablavt

22 + Ra nablaτt

22 minus e

minus βznablaΠt vt1113872 1113873 + 2Ra τt vzt( 1113857 +

minus1Pr

vt middot nablav + v middot nablavt vt( 1113857 minus1Pr

ϕtv + ϕvt vt( 1113857 minus Ra vt middot nablaτ + v middot nablaτt τt( 1113857

(75)

Technically the estimates of the nonlinear convectiveterms are the only different ones from the previous onessince the terms are not homogeneous

eminus βznablaΠt vt1113872 1113873

11138681113868111386811138681113868τt0le c vt

2 vt middot nablav

2 + v middot nablavt

21113872 1113873

le c vt

2 vt

3nablav6 + vt

2vinfin nablavt

21113872 1113873

le c vt

322 nablavt

122 Δv2 + vt

2Δv2 nablavt

21113874 1113875

le c Δv22 vt

22 + ε nablavt

221113874 1113875

(76)

e same estimates can be repeated for the other non-linear terms such as

ϕvt vt( 1113857le cnablav6 vt

2 vt

3

ϕtv vt( 1113857le c vt

2vinfin nablavt

2

vt middot nablaτ τt( 1113857le cnablaτ6 vt

2 τt

3

v middot nablaτt τt( 1113857le c τt

2vinfin nablaτt

2

(77)

e gradient terms can be absorbed at the left-handside

Starting from (74) where the positive term at the left-hand side can be neglected and the term vt

22 at the right-

International Journal of Differential Equations 9

hand side can be bounded by E(t) by a generalizedGronwall argument one shows that

E(t)le ec 1113938

T

0Δv(t)22+1( )dt

E(0) (78)

As a matter of fact (71) implies v isin L2(0 T W22(Ω0))so thatE(t) is bounded in (0 T) Further by coming back to(74) and integrating again it follows also

1113946T

0nablavt

22 + nablaτt

221113874 1113875dtleO v0

22 τ0

221113872 1113873 (79)

and this is independent of the size of Ra

Theorem 1 Let βlt 2π and Pr and Ra be arbitrary thensystem (25) with initial data (v0 τ0) isinW22(Ω0) admits asolution a la Ladyzhenskaya namely

(v τ) isin Linfin 0 T W

22 Ω0( 11138571113872 1113873capL2 0 T W

32 Ω0( 11138571113872 1113873

vt τt( 1113857 isin Linfin 0 T L

2 Ω0( 11138571113872 1113873cap L2 0 T W

12 Ω0( 11138571113872 1113873(80)

where

T ≔ logE2(0) + 1

E2(0)1113888 1113889

14C

(81)

Moreover if nabla middot v0 0 then nabla middot v(t) 0 for all t isin (0 T]Finally if Ra v022 and τ022 are sufficiently smalldepending on Pr and β then T infin is taken in (80) and (v τ)

decays exponentially fast to zero

Proof Employing the estimates proved in Lemma 4 it isroutine to show that the approximate solutions constructedwith the Galerkin method converge to a solution

(v τ) isin Linfin 0 T W

12 Ω0( 11138571113872 1113873capL2 0 T W

22 Ω0( 11138571113872 1113873

Π isin L2 0 T W

22 Ω0( 11138571113872 1113873

vt τt( 1113857 isin Linfin 0 T L

2 Ω0( 11138571113872 1113873cap L2 0 T W

12 Ω0( 11138571113872 1113873

(82)

where T is given in (81)e proof of the further regularity properties implying

that the solution fulfils system (17) too is based on (78) and(79) and in particular on the uniform boundedness of vt2as well as the integrability of nablavt(t)22 To this end viewingthe momentum balance as a Poisson problem and increasingthe pressure in the usual way the following is obtained withthe help of Lemma 1

v(t)22 leC v middot nablav2 +nablaΠ2 + vt

2 +τ21113872 1113873

leC nablav42 + εΔv

22 + vt

2 +τ21113872 1113873

(83)

e ε term at the right-hand side can be absorbed by theleft-hand side so that finally the following bound is obtained

ess sup(0T)

v(t)22 leCess sup(0T)

nablav(t)42 +τ2 + vt

21113872 1113873

leO v0

22 τ0

221113872 1113873

(84)

Moreover again from Lemma 1 it is also obtained that

1113946T

0v(t)

232dtleC 1113946

T

0v middot nablav

212 +Π

222 + vt

212 +τ

2121113874 1113875dt

(85)

It is remarked that as noticed in Section 2 the com-patibility condition related to the Neumann problem isverified Here the nonlinear terms at the right-hand side canbe increased by using (14) and interpolation inequality

v middot D2v

2 le cvinfin D

2v

2 le c D2v

22

nablav(nablav)T

le cnablav

24 le cnablav6nablav3 le c D

2v

22

(86)

e proof that τ is bounded in Linfin(0 T W12(Ω0))capL2(0 T W22(Ω0)) is similar and simpler

e proof of global existence follows from Lemmas 3 and5 by setting

E0(t) ≔ E(t) + E(t)

P0(t) ≔ nablav(t)22 +Δv(t)

22 + nablavt(t)

22

+ Ra nablaτ(t)22 +Δτ(t)

22 + nablaτt(t)

221113874 1113875

(87)

In fact if inequalities (49) (51) and (74) are addedbringing to the left-hand side the three terms with factorRaε and using the Poincare inequality

Ele kP (88)

to absorb them in P0 for sufficiently small Ra (dependingon ε) one gets

ddt

E(t) + c1P(t)le c2 E32

(t) + E3(t)1113872 1113873 (89)

e constant c1 is now fixed positive by suitablyrestricting Ra Again using the Poincare inequality on theleft-hand side of (89) one infers

ddt

E(t) + c1 minus c2 E12

(t) + E2(t)1113872 11138731113960 1113961

1k

E(t)le 0 (90)

us the exponential decay follows by choosing at theinitial time

E12

(0) + E2(0)le

12

c1

c2 (91)

In fact by continuity for some time E(t)leE(0) Butthis reinforces the previous condition so that the positivefunction in square brackets cannot decrease and is boundedfrom below by a positive constant for all times

Data Availability

e data that support the findings of this study are availablewithin the article

Conflicts of Interest

e author declares that there are no conflicts of interest

10 International Journal of Differential Equations

References

[1] A Giacobbe and G Mulone ldquoStability in the rotating Benardproblem and its optimal Lyapunov functionsrdquo Acta Appli-candae Mathematicae vol 132 no 1 2014

[2] A Passerini C Ferrario M Ruzicka and G ater ldquoe-oretical results on steady convective flows between horizontalcoaxial cylindersrdquo SIAM Journal on Applied Mathematicsvol 71 no 2 pp 465ndash486 2011

[3] A Passerini and C Ferrario ldquoA theoretical study of the firsttransition for the non-linear Stokes problem in a horizontalannulusrdquo International Journal of Non-linear Mechanicsvol 78 no 1 pp 1ndash8 2015

[4] A De Martino and A Passerini ldquoExistence and nonlinearstability of convective solutions for almost compressible fluidsin Benard problemrdquo Journal of Mathematical Physics vol 60no 11 Article ID 113101 2019

[5] F Crispo and P Maremonti ldquoA high regularity result ofsolutions to modified p-Stokes equationsrdquo Nonlinear Anal-ysis vol 118 pp 97ndash129 2015

[6] H Gouin and T Ruggeri ldquoA consistent thermodynamicalmodel of incompressible media as limit case of quasi-thermal-incompressible materialsrdquo International Journal of Non-Linear Mechanics vol 47 no 6 pp 688ndash693 2012

[7] A Passerini and T Ruggeri ldquoe Benard problem for quasi-thermal-incompressible materials a linear analysisrdquo Inter-national Journal of Non-Linear Mechanics vol 67 no 1pp 178ndash185 2014

[8] S Gatti and E Vuk ldquoSingular limit of equations for linearviscoelastic fluids with periodic boundary conditionsrdquo In-ternational Journal of Non-Linear Mechanics vol 41 no 4pp 518ndash526 2006

[9] A Corli and A Passerini ldquoe Benard problem for slightlycompressible materials existence and linear instabilityrdquoMediterranean Journal of Mathematics vol 16 no 1 2019

[10] G P Galdi ldquoNon-linear stability of the magnetic Benardproblem via a generalized energy methodrdquo Archive for Ra-tional Mechanics and Analysis vol 87 no 2 pp 167ndash1861985

[11] G P Galdi and M Padula ldquoA new approach to energy theoryin the stability of fluid motionrdquo Archive for Rational Me-chanics and Analysis vol 110 no 3 pp 187ndash286 1990

[12] A De Martino and A Passerini ldquoA Lorenz model for almostcompressible fluidsrdquo Mediterranean Journal of Mathematicsvol 17 no 1 p 12 2020

[13] G P Galdi An Introduction to the Mathematical eory of theNavier-Stokes Equation Springer New York NY USA 1998

[14] O Ladyzhenskaya e Mathematical eory of Viscous In-compressible Fluids Gordon amp Breach London UK 1963

[15] G Iooss ldquoeorie non lineaire de la stabilite des ecoulementslaminaires dans le cas de ldquolrsquoechange des stabilitesrdquo Archive forRational Mechanics and Analysis vol 40 no 3 pp 166ndash2081971

[16] M Schechter ldquoCoerciveness in Lprdquo Transactions of theAmerican Mathematical Society vol 107 no 1 pp 10ndash291963

International Journal of Differential Equations 11

is means that the divergence-free condition can be ful-filled and is moreover physically observable

Hereafter a solution with the required regularity isshown to exist provided the initial data for (19)2 and (19)3are in W22(Ω0)

Concerning boundary conditions the change of variableΠ Peβz turns Robinrsquos ones into the simpler Neumannconditions

Πz 0 z 0 1 (24)

and simplifies the system under study which becomes likethe one proposed in [9] page 7 and studied in [4]

ΔΠ minus βΠz minuseβz

Prnabla middot (v middot nablav + ϕv) + Rae

βzτz

1Pr

zvzt

+ v middot nablav + ϕv1113888 1113889 minus Δv minus nabla eminus βzΠ1113872 1113873 minus βe

minus βzΠk

+Raτk

zτzt

+ v middot nablaτ minus Δτ v middot k

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(25)

enew variableΠ is defined up to an arbitrary constantunlike P which appears in the buoyancy force and dependson the boundary value pb which is in the basic solutionus Π can be prescribed with a mean value zero and verifythe Poincare inequality

e trivial solution (v c1i + c2j τ 0 P 0) corre-sponding to the Galilean invariance has to be consideredequivalent to the rest state so for this reason solutions arelooked in Banach subspaces defined by the following condition

langvxrang langvyrang 0 (26)

Next one assigns in Ω0 the periodicity conditions in thehorizontal directions and the stress-free conditions at theboundaries and homogeneous conditions for the deviatorytemperature τ

Πz(x y 0 t) Πz(x y 1 t) 0

vz(x y 0 t) v

z(x y 1 t) 0

vxz(x y 0 t) v

xz(x y 1 t) 0

τ(x y 0 t) τ(x y 1 t) 0

(27)

At the initial time data are given concordant with theprevious ones

(v(x y z 0) τ(x y z 0)) v0(x y z) τ0(x y z)( 1113857

(28)

e plan is studying (17) before proving the equivalencewith (25) To this end one looks for solutions such that aein t and for k isin N0Π belongs to Wk2(Ω0) which is definedas the closure in the Sobolev space norm of Cinfin(Ω0) whosefunctions are periodic in x and y verifies the homogeneousNeumann condition at z 0 1 and has a zero mean value(this last feature is necessary to prove the existence)

Analogously without the isochoric conditionsv equiv (vx vy vz) has free components such that vx and vy

belong toWk2(Ω0) (with the mean value zero as previouslynoticed) while vz and τ belong to Wk2

0 (Ω0) which is theclosure of Cinfin0 (Ω0)

3 Galerkin Approximation

Still in [9] and in [4] an existence result for Π and furtherestimates are given in 2D e proof is unaffected by thedimension and the results are summarized below

Lemma 2 If 0lt βlt 2π and if f isin L2(Ω0) then the problem

ΔΠ minus βΠz eβzf inΩ0

Πz(x y 0) Πz(x y 1) 0 for (x y) isin R2

⎧⎨

⎩ (29)

with periodic side conditions has a unique solutionΠ isinW22(Ω0) such that langΠrang 0 and the following estimateholds true

Π22 leC(β)f2 (30)

In this lemma and hereafter C(β) stands for a positivefunction of β such that limβ⟶0C(β) cge 0 e sameconvention is kept for C(β PrRa)

Furthermore if f nabla middot w and w isin L2(Ω0) with w middot k 0at z 0 1 then in [4] one can also find

Π12 leC(β)w2 (31)

If moreover f nabla middot (nabla middot (uotimesu)) with u middot k 0 at z

0 1 and nabla middot u 0 then in [4] one can also find

Π2 leC(β)u24 (32)

Notice that both these estimates fall into the classicNavierndashStokes estimate

p2 le cv24 (33)

as β goes to zeroHowever here the case nabla middot v ne 0 is in principle

consideredBy taking the data of (29) from system (25) all hy-

potheses concerning the boundary conditions are verifiedand by writing

f minus1Prnabla middot (v middot nablav + ϕv) + Ranabla middot (τk) (34)

one sees that (31) can be usedHence

nablaΠ2 leC(β)1Pr

v middot nablav2 +ϕv2( 1113857 + Raτ21113874 1113875 (35)

Now let us introduce Galerkinrsquos ldquoapproximaterdquo solu-tions of (25) Let us denote by φj1113966 1113967

jisinN and χj1113966 1113967jisinN the

periodic eigenfunctions of the Laplace operator respectivelyverifying Dirichlet and Neumann conditions at zΩ suchthat the vector functions Ψj (χj χjφj) verify the sameboundary condition as v does

4 International Journal of Differential Equations

ΔΨj minus λ(j)Ψj λ(j) gt 0

Δφj minus η(j)φj η(j) gt 0(36)

Moreover Ψjrsquos and φjrsquos are complete orthogonal basesrespectively for the vector and scalar fields in W12(Ω0)Further they belong to W22(Ω0) e Galerkin approxi-mation solutions vN and τN

vN 1113944

N

j1C

jN(t)Ψj

τN 1113944

N

j1A

jN(t)φj

(37)

are then defined in a standard way by projecting system (25)on the N-dimensional subspace spanned by the sameeigenfunctions e ordinary differential equation system soobtained is the first order in the unknowns C

jN(t) and

AjN(t) It is an autonomous homogeneous system in a

normal form whose solutions are easily shown to exist for allN isin N

As a matter of fact although the pressure terms in themomentum balance (which will be written in a compactform as eminus βznablaΠ) cannot be solved a posteriori as forNavierndashStokes equations one can nevertheless define for allN isin N some suitable ΠN as a function of C

j

N(t) and Aj

N(t)To this end let us use (34) in Lemma 2 and denote byA thelinear operator

A ΠN1113872 1113873 ≔ ΔΠN

minus βΠNz e

βznabla

middot minus1Pr

vNmiddot nablavN

+ ϕNvN1113872 1113873 + RaτNk1113874 1113875

(38)

where of course ϕN nabla middot vN By Lemma 2 the operator iscontinuously invertible for 0lt βlt 2π and

ΠN ≔ minus1Pr

1113944

N

ki1A

minus 1eβznabla middot Ψk middot nablaΨi + Ψinabla middot Ψk( 11138571113872 1113873C

kN(t)C

iN(t)

+ RaAminus 11113944

N

i1eβznabla middot φik( 11138571113872 1113873A

iN(t)

(39)

where the summation over repeated indices is understoodAccording to Lemma 2 the linear operatorAminus 1 is defined in

L2(Ω0) and its image belongs to W22(Ω0) Moreover Aminus 1

does not depend on tFor the sake of brevity let us denote

Fki(x y z) ≔ Aminus 1

eβznabla middot Ψk middot nablaΨi + Ψinabla middot Ψk( 11138571113872 1113873

Gi(x y z) ≔ Aminus 1

eβznabla middot φik( 11138571113872 1113873

(40)

Next let us compute

eminus βznablaΠN

Ψj1113872 1113873 minus1Pr

1113944

N

ki1e

minus βznablaFkiΨj1113872 1113873CkNC

iN

+ Ra1113944

N

i1e

minus βznablaGiΨj1113872 1113873AiN

(41)

is expression can be inserted in the Galerkin systemwhich can be written as

_Cj

N minus Prλ(j)CjN minus 1113944

N

ki1Ψk middot nablaΨi minus e

minus βznablaFkiΨj1113872 1113873CkNC

iN+

+ PrRa1113944N

i1φik minus e

minus βznablaGiΨj1113872 1113873AiN

(42)

_Aj

N minus 1113944N

ki1C

kNA

iN Ψk middot nablaφiφj1113872 1113873 minus η(j)A

j

N + Cj

N (43)

It allows for solutions since the right-hand side is aLipschitz continuous function of the variables (in fact it is atmost quadratic) For all N isin N the initial conditionsC

jN(0) C

j0 (v0Ψj) and A

jN(0) A

j0 (τ0Ψj) corre-

spond to the initial conditions of (25) Since the aim is still toderive the equivalence of the differential systems (17) and(19) by showing existence with ϕ 0 the required regularityfor the initial data is W22 In particular one has to define thespace as the closure of linear combinations of the eigen-functions χj minus langχjrang so that vx vy isinW22(Ω0) whilevz τ isinW22

0 (Ω0) which is directly obtained by closing theφjrsquos

Moreover an estimate for vNt (0)2 and τN

t (0)2 ishereafter needed for which the left-hand side of (42)evaluated in t 0 has to be increased by directly inserting(34) in estimate (31)

eminus βznablaΠN

Ψj1113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868le eminus β nablaΠN

Ψj1113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868

leC(β PrRa) vNmiddot nablavN

+ ϕNvNΨj1113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868 + τNΨj1113872 1113873

11138681113868111386811138681113868

111386811138681113868111386811138681113874 1113875

leC(β PrRa) 1113944N

ki1C

kNC

iN Ψk middot nablaΨi + Ψinabla middot ΨkΨj1113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868 + AjN

11138681113868111386811138681113868

11138681113868111386811138681113868⎛⎝ ⎞⎠

(44)

International Journal of Differential Equations 5

After squaring and summing the inequalities derived by(42) and evaluating at the initial time one obtains a boundindependent of N estimated by the W22-norm of the initial

data through Besselrsquos inequality Since the basis is inW22(Ω0) by linearity

ΔvN(0)

2 1113944

N

j1λ(j) v0Ψj1113872 1113873Ψj

2

1113944N

j1v0ΔΨj1113872 1113873Ψj

2

1113944N

j1Δv0Ψj1113872 1113873Ψj

2

le Δv0

2 (45)

Concerning the nonlinear terms one can compute forinstance

1113944

N

j11113944

N

ki11113944

N

ln1Ψk middot nablaΨiΨj1113872 1113873 Ψl middot nablaΨnΨj1113872 1113873C

kN(0)C

iN(0)C

lN(0)C

nN(0)

le 1113944

infin

j1v0 middot nablav0Ψj1113872 1113873

2 v0 middot nablav0

2

(46)

Next these nonlinear terms at the right-hand side can beincreased by (12) and (13) through Holder (p (32) andpprime 3) and interpolation (p (43) and pprime 4) inequalities(the procedure is detailed in the proof of Lemma 3) Finallyusing the Poincare inequality and (10) one gets

vNt (0)

2 leC(β PrRa) Δv0

2 + τ0

2 + Δv0

221113874 1113875

τNt (0)

2 leC(β PrRa) Δτ0

2 + v0

2 + nablav0

2 Δτ0

21113872 1113873

(47)

Here and hereafter the L2-norms of the two nonlinearterms v middot nablav and ϕv are estimated exactly in the same waybut for brevity only the first one will be developed

4 Evolutive Estimates and Main Results

Let us formally derive the a priori estimates for solutions ofsystem (25) by quite a standard procedure they are verifiedjust by Galerkin solutions denoted here by (v τ) and asseen are bound uniformly with respect to N

Lemma 3 Let us define the energy functions

E1(t) ≔12

v22Pr

+ Raτ221113888 1113889

E2(t) ≔12

nablav22Pr

+ Ranablaτ221113888 1113889

(48)

en for sufficiently small εgt 0 the following inequalitieshold true

dE1

dt+ c nablav

22 + Ranablaτ

221113872 1113873leC(β PrRa) E

31 + E

322 + εΔv

221113872 1113873 + C(β Pr)

Raε

v22 (49)

dE2

dt+ c Δv

22 + RaΔτ

221113872 1113873leC(β PrRa)E

32 + C(β Pr)

Raε

E1 (50)

dE2

dt+ c Δv

22 + Raτ

221113872 1113873leC(β PrRa)E

32 + C(β Pr)

Raε

Δv22 (51)

where c isin (0 1) Proof One can test (25) with (v τ) and then one multipliesthe third equation by Ra and then sums

ddt

E1(t) +nablav22 + Ranablaτ

22 minus e

minus βznablaΠ v1113872 1113873 + 2Ra τ vz

( 1113857

minus1Pr

(v middot nablav v) minus1Pr

(ϕv v) minus Ra(v middot nablaτ τ)

(52)

6 International Journal of Differential Equations

In order to estimate the nonlinear term arising from thecoupling with the pressure equation one uses (35) thenonlinear term at the right-hand side can be increased by

(12) and (13) through Holder (p (32) and pprime 3) andinterpolation (p (43) and pprime 4) inequalities so that

eminus βznablaΠ v1113872 1113873

11138681113868111386811138681113868τ0le cv middot nablav2v2 le cv3nablav6v2 le cv

122 v

126 Δv2v2

le cv322 nablav

122 Δv2 le

c

4ε2v

62 +

cε4

nablav22 +

cε2

Δv22

(53)

is term is multiplied by β(C(β)Pr) while the linearpart of the pressure estimate can be included in the estimateof the buoyancy term and of the energy equation transportterm giving rise (in the whole) to

RaC(β Pr)τ2v2 leRaC(β Pr)ε2τ

22 +

12ε

v221113874 1113875 (54)

Here the termwith the second derivatives does not allowto get an autonomous bound for E1(t) so that a furtherestimate for higher derivatives need to be added Notice thatthe convective terms do not vanish and by also resorting toYoungrsquos inequality they are bounded as follows

(v middot nablav v) 1113946Ω0nabla middot v

v2

21113888 1113889 minus 1113946

Ω0ϕ

v2

2le12ϕ2v

24

leCnablav2v122 v

326 leCnablav

32

(v middot nablaτ τ) 1113946Ω0nabla middot v

τ2

21113888 1113889 minus 1113946

Ω0ϕτ2

2le12ϕ2τ

24

leCnablav2τ122 τ

326 leCnablav2nablaτ

22 le

13

Cnablav32 +

23

Cnablaτ32

(55)

Analogously

1113946Ω0ϕv

2 le nablav32 (56)

By collecting all these estimates using the Poincareinequality increasing nablav32 with E32

2 and absorbing at the

left-hand side where possible the terms of order ϵ the firstinequality follows

Now the balance equations are tested with Δv and Δτrespectively and again (35) is inserted

First the following equations are written

12Pr

ddt

nablav22 +Δv

22

1Pr

(v middot nablav + ϕvΔv) + eminus βznablaΠΔv1113872 1113873 minus Ra(τkΔv)

12ddt

nablaτ22 +Δτ

22 (v middot nablaτΔτ) + v

zΔτ( 1113857

dE2

dt+Δv

22 + RaΔτ

22 e

minus βznablaΠΔv1113872 1113873 + Ra τΔvz( 1113857 + Ra Δτ v

z( 1113857

1Pr

(v middot nablavΔv) +1Pr

(ϕvΔv) + Ra(v middot nablaτΔτ)

(57)

Again the pressure-related convective terms arebounded and then the temperature-depending couplingterms are estimated

eminus βznablaΠΔv1113872 1113873

11138681113868111386811138681113868τ0le cv middot nablav2Δv2 le v6nablav3Δv2 le

cv6nablav122 D

2v

122 Δv2 le cnablav

322 Δv

322 le

3cε4

Δv22 +

c

4εnablav

62

(58)

International Journal of Differential Equations 7

In order to reach local-in-time existence of solutions forany initial data one writes

RaC(β Pr) τ2Δv2 +Δτ2v2( 1113857leRaC(β Pr)ε2Δτ

22 +

ε2Δv

22 +

12ε2

E1(t)1113874 1113875 (59)

In order to achieve global-in-time existence for smallinitial data one uses an alternative estimate which followsby several applications of the Poincare inequality

RaC(β Pr) τ2Δv2 +Δτ2v2( 1113857

leRaC(β Pr)ε2Δτ

22 +

12ε

Δv221113874 1113875

(60)

Now the nonlinear terms are focused

(v middot nablav + ϕvΔv)leCv6nablav3Δv2 leCv6nablav122 nablav

126 Δv2

leCnablav322 Δv

322 le

3ε4

CΔv22 +

14ε

Cnablav62

(v middot nablaτΔτ)le v6nablaτ3Δτ2 leCnablav2nablaτ122 D

2τ

122 Δτ2

leCnablav2nablaτ122 Δτ

322 le

3ε4

CΔτ22 +

14ε

Cnablaτ22nablav

42

le3ε4

CΔτ22 +

112ε

Cnablaτ62 +

16ε

Cnablav62

(61)

e terms of order ϵ are still absorbed at the left-handside and then inequalities (50) and (51) respectively followby (59) and (60)

Lemma 4 Let

T ≔ logE2(0) + 1

E2(0)1113888 1113889

14C

(62)

then for all t isin (0 T] the following estimates hold

E(t)le E(0)

1 + E2(0)

1113968

1 + E(0) E(0) minus1 + E2(0)

11139681113872 1113873

⎛⎝ ⎞⎠

12

(63)

where E(t) ≔ E1(t) + E2(t) Furthermore

1113946T

0P(t)dtleC log

E2(0) + 1E2(0)

1113888 1113889

14

E(0) + E12

(0)1113872 1113873

(64)

where P(t) ≔ nablav(t)22 + Δv(t)22 + nablaτ(t)22 + Δτ(t)22

Proof By summing side-by-side the first and second in-equalities in Lemma 3 one obtains

ddt

E(t) + c1P(t)le c2 E(t) + E32

(t) + E3(t)1113872 1113873 (65)

Here one has

E32

E34

E34 le

34

E +14E3 (66)

and consequently the right-hand side can be bounded byE + E3 Hence

1113946E(t)

E(0)

dE

E E2 + 1( ) 1113946

E(t)

E(0)

1E

minusE

E2 + 11113874 1113875dEleCt (67)

then

E(t)E2(0) + 1

1113968

E(0)E2(t) + 1

1113968 le eCt

(68)

It follows that once the maximal interval correspondingto a vanishing denominator is identified

forallt isin (0 T)with 2T ≔ logE2(0) + 1

E2(0)1113888 1113889

12C

(69)

E2(t)le

e2CtE2(0)

1 + E2(0) 1 minus e2Ct( )

leE(0)

1 + E2(0)

1113968

1 + E(0) E(0) minus1 + E2(0)

11139681113872 1113873

(70)

8 International Journal of Differential Equations

the energy is boundedNext coming back to (65) and integrating in [0 T) one

deduces a bound for the norm in Linfin(0 T W12(Ω0))capL2(0 T W22(Ω0)) as can be seen hereafter the inequality

1113946T

0P(t)dtleC log

E2(0) + 1E2(0)

1113888 1113889

14

E(0) + E12

(0)1113872 1113873

(71)

implies in particular the integrability of nablav(t)22 and ofΔv(t)22

Although the Galerkin solutions exist for all tgt 0 theiruniform estimates exist only in the finite interval defined in(69) Global-in-time solution will be found only by askingsuitable smallness of Ra and of the initial data

Finally (25) is derived formally with respect to t and thesame procedure is repeated as in the first estimate but for thenonlinear convective terms Now in order to bound theinitial data for vt initial data in W22 is chosen

ΔΠt minus βΠtz minuseβz

Prnabla middot vt middot nablav + ϕtv + v middot nablavt + ϕvt( 1113857

+Raeβzτtz

1Pr

zvt

zt+ vt middot nablav + ϕtv + v middot nablavt + ϕvt1113888 1113889 minus Δvt

minus nabla eminus βzΠt( 1113857 minus βeminus βzΠtk + Raτtk

zτt

zt+ vt middot nablaτ + v middot nablaτt minus Δτt v

zt

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(72)

Again the terms depending on ϕ or ϕt will be boundedexactly as the corresponding convective ones

Lemma 5 Let us define the energy function as

E(t) ≔12

vt

22

Pr+ Ra τt

22

⎛⎝ ⎞⎠ (73)

en for sufficiently small εgt 0 the following inequalityholds truedEdt

(t) + c nablavt

22 + Ra nablaτt

221113874 1113875leC(β PrRa)Δv

22E(t)

+ C(βPr)Raε

vt

22

(74)

where c isin (0 1)

Proof Let us start with the equation

ddt

E(t) + nablavt

22 + Ra nablaτt

22 minus e

minus βznablaΠt vt1113872 1113873 + 2Ra τt vzt( 1113857 +

minus1Pr

vt middot nablav + v middot nablavt vt( 1113857 minus1Pr

ϕtv + ϕvt vt( 1113857 minus Ra vt middot nablaτ + v middot nablaτt τt( 1113857

(75)

Technically the estimates of the nonlinear convectiveterms are the only different ones from the previous onessince the terms are not homogeneous

eminus βznablaΠt vt1113872 1113873

11138681113868111386811138681113868τt0le c vt

2 vt middot nablav

2 + v middot nablavt

21113872 1113873

le c vt

2 vt

3nablav6 + vt

2vinfin nablavt

21113872 1113873

le c vt

322 nablavt

122 Δv2 + vt

2Δv2 nablavt

21113874 1113875

le c Δv22 vt

22 + ε nablavt

221113874 1113875

(76)

e same estimates can be repeated for the other non-linear terms such as

ϕvt vt( 1113857le cnablav6 vt

2 vt

3

ϕtv vt( 1113857le c vt

2vinfin nablavt

2

vt middot nablaτ τt( 1113857le cnablaτ6 vt

2 τt

3

v middot nablaτt τt( 1113857le c τt

2vinfin nablaτt

2

(77)

e gradient terms can be absorbed at the left-handside

Starting from (74) where the positive term at the left-hand side can be neglected and the term vt

22 at the right-

International Journal of Differential Equations 9

hand side can be bounded by E(t) by a generalizedGronwall argument one shows that

E(t)le ec 1113938

T

0Δv(t)22+1( )dt

E(0) (78)

As a matter of fact (71) implies v isin L2(0 T W22(Ω0))so thatE(t) is bounded in (0 T) Further by coming back to(74) and integrating again it follows also

1113946T

0nablavt

22 + nablaτt

221113874 1113875dtleO v0

22 τ0

221113872 1113873 (79)

and this is independent of the size of Ra

Theorem 1 Let βlt 2π and Pr and Ra be arbitrary thensystem (25) with initial data (v0 τ0) isinW22(Ω0) admits asolution a la Ladyzhenskaya namely

(v τ) isin Linfin 0 T W

22 Ω0( 11138571113872 1113873capL2 0 T W

32 Ω0( 11138571113872 1113873

vt τt( 1113857 isin Linfin 0 T L

2 Ω0( 11138571113872 1113873cap L2 0 T W

12 Ω0( 11138571113872 1113873(80)

where

T ≔ logE2(0) + 1

E2(0)1113888 1113889

14C

(81)

Moreover if nabla middot v0 0 then nabla middot v(t) 0 for all t isin (0 T]Finally if Ra v022 and τ022 are sufficiently smalldepending on Pr and β then T infin is taken in (80) and (v τ)

decays exponentially fast to zero

Proof Employing the estimates proved in Lemma 4 it isroutine to show that the approximate solutions constructedwith the Galerkin method converge to a solution

(v τ) isin Linfin 0 T W

12 Ω0( 11138571113872 1113873capL2 0 T W

22 Ω0( 11138571113872 1113873

Π isin L2 0 T W

22 Ω0( 11138571113872 1113873

vt τt( 1113857 isin Linfin 0 T L

2 Ω0( 11138571113872 1113873cap L2 0 T W

12 Ω0( 11138571113872 1113873

(82)

where T is given in (81)e proof of the further regularity properties implying

that the solution fulfils system (17) too is based on (78) and(79) and in particular on the uniform boundedness of vt2as well as the integrability of nablavt(t)22 To this end viewingthe momentum balance as a Poisson problem and increasingthe pressure in the usual way the following is obtained withthe help of Lemma 1

v(t)22 leC v middot nablav2 +nablaΠ2 + vt

2 +τ21113872 1113873

leC nablav42 + εΔv

22 + vt

2 +τ21113872 1113873

(83)

e ε term at the right-hand side can be absorbed by theleft-hand side so that finally the following bound is obtained

ess sup(0T)

v(t)22 leCess sup(0T)

nablav(t)42 +τ2 + vt

21113872 1113873

leO v0

22 τ0

221113872 1113873

(84)

Moreover again from Lemma 1 it is also obtained that

1113946T

0v(t)

232dtleC 1113946

T

0v middot nablav

212 +Π

222 + vt

212 +τ

2121113874 1113875dt

(85)

It is remarked that as noticed in Section 2 the com-patibility condition related to the Neumann problem isverified Here the nonlinear terms at the right-hand side canbe increased by using (14) and interpolation inequality

v middot D2v

2 le cvinfin D

2v

2 le c D2v

22

nablav(nablav)T

le cnablav

24 le cnablav6nablav3 le c D

2v

22

(86)

e proof that τ is bounded in Linfin(0 T W12(Ω0))capL2(0 T W22(Ω0)) is similar and simpler

e proof of global existence follows from Lemmas 3 and5 by setting

E0(t) ≔ E(t) + E(t)

P0(t) ≔ nablav(t)22 +Δv(t)

22 + nablavt(t)

22

+ Ra nablaτ(t)22 +Δτ(t)

22 + nablaτt(t)

221113874 1113875

(87)

In fact if inequalities (49) (51) and (74) are addedbringing to the left-hand side the three terms with factorRaε and using the Poincare inequality

Ele kP (88)

to absorb them in P0 for sufficiently small Ra (dependingon ε) one gets

ddt

E(t) + c1P(t)le c2 E32

(t) + E3(t)1113872 1113873 (89)

e constant c1 is now fixed positive by suitablyrestricting Ra Again using the Poincare inequality on theleft-hand side of (89) one infers

ddt

E(t) + c1 minus c2 E12

(t) + E2(t)1113872 11138731113960 1113961

1k

E(t)le 0 (90)

us the exponential decay follows by choosing at theinitial time

E12

(0) + E2(0)le

12

c1

c2 (91)

In fact by continuity for some time E(t)leE(0) Butthis reinforces the previous condition so that the positivefunction in square brackets cannot decrease and is boundedfrom below by a positive constant for all times

Data Availability

e data that support the findings of this study are availablewithin the article

Conflicts of Interest

e author declares that there are no conflicts of interest

10 International Journal of Differential Equations

References

[1] A Giacobbe and G Mulone ldquoStability in the rotating Benardproblem and its optimal Lyapunov functionsrdquo Acta Appli-candae Mathematicae vol 132 no 1 2014

[2] A Passerini C Ferrario M Ruzicka and G ater ldquoe-oretical results on steady convective flows between horizontalcoaxial cylindersrdquo SIAM Journal on Applied Mathematicsvol 71 no 2 pp 465ndash486 2011

[3] A Passerini and C Ferrario ldquoA theoretical study of the firsttransition for the non-linear Stokes problem in a horizontalannulusrdquo International Journal of Non-linear Mechanicsvol 78 no 1 pp 1ndash8 2015

[4] A De Martino and A Passerini ldquoExistence and nonlinearstability of convective solutions for almost compressible fluidsin Benard problemrdquo Journal of Mathematical Physics vol 60no 11 Article ID 113101 2019

[5] F Crispo and P Maremonti ldquoA high regularity result ofsolutions to modified p-Stokes equationsrdquo Nonlinear Anal-ysis vol 118 pp 97ndash129 2015

[6] H Gouin and T Ruggeri ldquoA consistent thermodynamicalmodel of incompressible media as limit case of quasi-thermal-incompressible materialsrdquo International Journal of Non-Linear Mechanics vol 47 no 6 pp 688ndash693 2012

[7] A Passerini and T Ruggeri ldquoe Benard problem for quasi-thermal-incompressible materials a linear analysisrdquo Inter-national Journal of Non-Linear Mechanics vol 67 no 1pp 178ndash185 2014

[8] S Gatti and E Vuk ldquoSingular limit of equations for linearviscoelastic fluids with periodic boundary conditionsrdquo In-ternational Journal of Non-Linear Mechanics vol 41 no 4pp 518ndash526 2006

[9] A Corli and A Passerini ldquoe Benard problem for slightlycompressible materials existence and linear instabilityrdquoMediterranean Journal of Mathematics vol 16 no 1 2019

[10] G P Galdi ldquoNon-linear stability of the magnetic Benardproblem via a generalized energy methodrdquo Archive for Ra-tional Mechanics and Analysis vol 87 no 2 pp 167ndash1861985

[11] G P Galdi and M Padula ldquoA new approach to energy theoryin the stability of fluid motionrdquo Archive for Rational Me-chanics and Analysis vol 110 no 3 pp 187ndash286 1990

[12] A De Martino and A Passerini ldquoA Lorenz model for almostcompressible fluidsrdquo Mediterranean Journal of Mathematicsvol 17 no 1 p 12 2020

[13] G P Galdi An Introduction to the Mathematical eory of theNavier-Stokes Equation Springer New York NY USA 1998

[14] O Ladyzhenskaya e Mathematical eory of Viscous In-compressible Fluids Gordon amp Breach London UK 1963

[15] G Iooss ldquoeorie non lineaire de la stabilite des ecoulementslaminaires dans le cas de ldquolrsquoechange des stabilitesrdquo Archive forRational Mechanics and Analysis vol 40 no 3 pp 166ndash2081971

[16] M Schechter ldquoCoerciveness in Lprdquo Transactions of theAmerican Mathematical Society vol 107 no 1 pp 10ndash291963

International Journal of Differential Equations 11

ΔΨj minus λ(j)Ψj λ(j) gt 0

Δφj minus η(j)φj η(j) gt 0(36)

Moreover Ψjrsquos and φjrsquos are complete orthogonal basesrespectively for the vector and scalar fields in W12(Ω0)Further they belong to W22(Ω0) e Galerkin approxi-mation solutions vN and τN

vN 1113944

N

j1C

jN(t)Ψj

τN 1113944

N

j1A

jN(t)φj

(37)

are then defined in a standard way by projecting system (25)on the N-dimensional subspace spanned by the sameeigenfunctions e ordinary differential equation system soobtained is the first order in the unknowns C

jN(t) and

AjN(t) It is an autonomous homogeneous system in a

normal form whose solutions are easily shown to exist for allN isin N

As a matter of fact although the pressure terms in themomentum balance (which will be written in a compactform as eminus βznablaΠ) cannot be solved a posteriori as forNavierndashStokes equations one can nevertheless define for allN isin N some suitable ΠN as a function of C

j

N(t) and Aj

N(t)To this end let us use (34) in Lemma 2 and denote byA thelinear operator

A ΠN1113872 1113873 ≔ ΔΠN

minus βΠNz e

βznabla

middot minus1Pr

vNmiddot nablavN

+ ϕNvN1113872 1113873 + RaτNk1113874 1113875

(38)

where of course ϕN nabla middot vN By Lemma 2 the operator iscontinuously invertible for 0lt βlt 2π and

ΠN ≔ minus1Pr

1113944

N

ki1A

minus 1eβznabla middot Ψk middot nablaΨi + Ψinabla middot Ψk( 11138571113872 1113873C

kN(t)C

iN(t)

+ RaAminus 11113944

N

i1eβznabla middot φik( 11138571113872 1113873A

iN(t)

(39)

where the summation over repeated indices is understoodAccording to Lemma 2 the linear operatorAminus 1 is defined in

L2(Ω0) and its image belongs to W22(Ω0) Moreover Aminus 1

does not depend on tFor the sake of brevity let us denote

Fki(x y z) ≔ Aminus 1

eβznabla middot Ψk middot nablaΨi + Ψinabla middot Ψk( 11138571113872 1113873

Gi(x y z) ≔ Aminus 1

eβznabla middot φik( 11138571113872 1113873

(40)

Next let us compute

eminus βznablaΠN

Ψj1113872 1113873 minus1Pr

1113944

N

ki1e

minus βznablaFkiΨj1113872 1113873CkNC

iN

+ Ra1113944

N

i1e

minus βznablaGiΨj1113872 1113873AiN

(41)

is expression can be inserted in the Galerkin systemwhich can be written as

_Cj

N minus Prλ(j)CjN minus 1113944

N

ki1Ψk middot nablaΨi minus e

minus βznablaFkiΨj1113872 1113873CkNC

iN+

+ PrRa1113944N

i1φik minus e

minus βznablaGiΨj1113872 1113873AiN

(42)

_Aj

N minus 1113944N

ki1C

kNA

iN Ψk middot nablaφiφj1113872 1113873 minus η(j)A

j

N + Cj

N (43)

It allows for solutions since the right-hand side is aLipschitz continuous function of the variables (in fact it is atmost quadratic) For all N isin N the initial conditionsC

jN(0) C

j0 (v0Ψj) and A

jN(0) A

j0 (τ0Ψj) corre-

spond to the initial conditions of (25) Since the aim is still toderive the equivalence of the differential systems (17) and(19) by showing existence with ϕ 0 the required regularityfor the initial data is W22 In particular one has to define thespace as the closure of linear combinations of the eigen-functions χj minus langχjrang so that vx vy isinW22(Ω0) whilevz τ isinW22

0 (Ω0) which is directly obtained by closing theφjrsquos

Moreover an estimate for vNt (0)2 and τN

t (0)2 ishereafter needed for which the left-hand side of (42)evaluated in t 0 has to be increased by directly inserting(34) in estimate (31)

eminus βznablaΠN

Ψj1113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868le eminus β nablaΠN

Ψj1113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868

leC(β PrRa) vNmiddot nablavN

+ ϕNvNΨj1113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868 + τNΨj1113872 1113873

11138681113868111386811138681113868

111386811138681113868111386811138681113874 1113875

leC(β PrRa) 1113944N

ki1C

kNC

iN Ψk middot nablaΨi + Ψinabla middot ΨkΨj1113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868 + AjN

11138681113868111386811138681113868

11138681113868111386811138681113868⎛⎝ ⎞⎠

(44)

International Journal of Differential Equations 5

After squaring and summing the inequalities derived by(42) and evaluating at the initial time one obtains a boundindependent of N estimated by the W22-norm of the initial

data through Besselrsquos inequality Since the basis is inW22(Ω0) by linearity

ΔvN(0)

2 1113944

N

j1λ(j) v0Ψj1113872 1113873Ψj

2

1113944N

j1v0ΔΨj1113872 1113873Ψj

2

1113944N

j1Δv0Ψj1113872 1113873Ψj

2

le Δv0

2 (45)

Concerning the nonlinear terms one can compute forinstance

1113944

N

j11113944

N

ki11113944

N

ln1Ψk middot nablaΨiΨj1113872 1113873 Ψl middot nablaΨnΨj1113872 1113873C

kN(0)C

iN(0)C

lN(0)C

nN(0)

le 1113944

infin

j1v0 middot nablav0Ψj1113872 1113873

2 v0 middot nablav0

2

(46)

Next these nonlinear terms at the right-hand side can beincreased by (12) and (13) through Holder (p (32) andpprime 3) and interpolation (p (43) and pprime 4) inequalities(the procedure is detailed in the proof of Lemma 3) Finallyusing the Poincare inequality and (10) one gets

vNt (0)

2 leC(β PrRa) Δv0

2 + τ0

2 + Δv0

221113874 1113875

τNt (0)

2 leC(β PrRa) Δτ0

2 + v0

2 + nablav0

2 Δτ0

21113872 1113873

(47)

Here and hereafter the L2-norms of the two nonlinearterms v middot nablav and ϕv are estimated exactly in the same waybut for brevity only the first one will be developed

4 Evolutive Estimates and Main Results

Let us formally derive the a priori estimates for solutions ofsystem (25) by quite a standard procedure they are verifiedjust by Galerkin solutions denoted here by (v τ) and asseen are bound uniformly with respect to N

Lemma 3 Let us define the energy functions

E1(t) ≔12

v22Pr

+ Raτ221113888 1113889

E2(t) ≔12

nablav22Pr

+ Ranablaτ221113888 1113889

(48)

en for sufficiently small εgt 0 the following inequalitieshold true

dE1

dt+ c nablav

22 + Ranablaτ

221113872 1113873leC(β PrRa) E

31 + E

322 + εΔv

221113872 1113873 + C(β Pr)

Raε

v22 (49)

dE2

dt+ c Δv

22 + RaΔτ

221113872 1113873leC(β PrRa)E

32 + C(β Pr)

Raε

E1 (50)

dE2

dt+ c Δv

22 + Raτ

221113872 1113873leC(β PrRa)E

32 + C(β Pr)

Raε

Δv22 (51)

where c isin (0 1) Proof One can test (25) with (v τ) and then one multipliesthe third equation by Ra and then sums

ddt

E1(t) +nablav22 + Ranablaτ

22 minus e

minus βznablaΠ v1113872 1113873 + 2Ra τ vz

( 1113857

minus1Pr

(v middot nablav v) minus1Pr

(ϕv v) minus Ra(v middot nablaτ τ)

(52)

6 International Journal of Differential Equations

In order to estimate the nonlinear term arising from thecoupling with the pressure equation one uses (35) thenonlinear term at the right-hand side can be increased by

(12) and (13) through Holder (p (32) and pprime 3) andinterpolation (p (43) and pprime 4) inequalities so that

eminus βznablaΠ v1113872 1113873

11138681113868111386811138681113868τ0le cv middot nablav2v2 le cv3nablav6v2 le cv

122 v

126 Δv2v2

le cv322 nablav

122 Δv2 le

c

4ε2v

62 +

cε4

nablav22 +

cε2

Δv22

(53)

is term is multiplied by β(C(β)Pr) while the linearpart of the pressure estimate can be included in the estimateof the buoyancy term and of the energy equation transportterm giving rise (in the whole) to

RaC(β Pr)τ2v2 leRaC(β Pr)ε2τ

22 +

12ε

v221113874 1113875 (54)

Here the termwith the second derivatives does not allowto get an autonomous bound for E1(t) so that a furtherestimate for higher derivatives need to be added Notice thatthe convective terms do not vanish and by also resorting toYoungrsquos inequality they are bounded as follows

(v middot nablav v) 1113946Ω0nabla middot v

v2

21113888 1113889 minus 1113946

Ω0ϕ

v2

2le12ϕ2v

24

leCnablav2v122 v

326 leCnablav

32

(v middot nablaτ τ) 1113946Ω0nabla middot v

τ2

21113888 1113889 minus 1113946

Ω0ϕτ2

2le12ϕ2τ

24

leCnablav2τ122 τ

326 leCnablav2nablaτ

22 le

13

Cnablav32 +

23

Cnablaτ32

(55)

Analogously

1113946Ω0ϕv

2 le nablav32 (56)

By collecting all these estimates using the Poincareinequality increasing nablav32 with E32

2 and absorbing at the

left-hand side where possible the terms of order ϵ the firstinequality follows

Now the balance equations are tested with Δv and Δτrespectively and again (35) is inserted

First the following equations are written

12Pr

ddt

nablav22 +Δv

22

1Pr

(v middot nablav + ϕvΔv) + eminus βznablaΠΔv1113872 1113873 minus Ra(τkΔv)

12ddt

nablaτ22 +Δτ

22 (v middot nablaτΔτ) + v

zΔτ( 1113857

dE2

dt+Δv

22 + RaΔτ

22 e

minus βznablaΠΔv1113872 1113873 + Ra τΔvz( 1113857 + Ra Δτ v

z( 1113857

1Pr

(v middot nablavΔv) +1Pr

(ϕvΔv) + Ra(v middot nablaτΔτ)

(57)

Again the pressure-related convective terms arebounded and then the temperature-depending couplingterms are estimated

eminus βznablaΠΔv1113872 1113873

11138681113868111386811138681113868τ0le cv middot nablav2Δv2 le v6nablav3Δv2 le

cv6nablav122 D

2v

122 Δv2 le cnablav

322 Δv

322 le

3cε4

Δv22 +

c

4εnablav

62

(58)

International Journal of Differential Equations 7

In order to reach local-in-time existence of solutions forany initial data one writes

RaC(β Pr) τ2Δv2 +Δτ2v2( 1113857leRaC(β Pr)ε2Δτ

22 +

ε2Δv

22 +

12ε2

E1(t)1113874 1113875 (59)

In order to achieve global-in-time existence for smallinitial data one uses an alternative estimate which followsby several applications of the Poincare inequality

RaC(β Pr) τ2Δv2 +Δτ2v2( 1113857

leRaC(β Pr)ε2Δτ

22 +

12ε

Δv221113874 1113875

(60)

Now the nonlinear terms are focused

(v middot nablav + ϕvΔv)leCv6nablav3Δv2 leCv6nablav122 nablav

126 Δv2

leCnablav322 Δv

322 le

3ε4

CΔv22 +

14ε

Cnablav62

(v middot nablaτΔτ)le v6nablaτ3Δτ2 leCnablav2nablaτ122 D

2τ

122 Δτ2

leCnablav2nablaτ122 Δτ

322 le

3ε4

CΔτ22 +

14ε

Cnablaτ22nablav

42

le3ε4

CΔτ22 +

112ε

Cnablaτ62 +

16ε

Cnablav62

(61)

e terms of order ϵ are still absorbed at the left-handside and then inequalities (50) and (51) respectively followby (59) and (60)

Lemma 4 Let

T ≔ logE2(0) + 1

E2(0)1113888 1113889

14C

(62)

then for all t isin (0 T] the following estimates hold

E(t)le E(0)

1 + E2(0)

1113968

1 + E(0) E(0) minus1 + E2(0)

11139681113872 1113873

⎛⎝ ⎞⎠

12

(63)

where E(t) ≔ E1(t) + E2(t) Furthermore

1113946T

0P(t)dtleC log

E2(0) + 1E2(0)

1113888 1113889

14

E(0) + E12

(0)1113872 1113873

(64)

where P(t) ≔ nablav(t)22 + Δv(t)22 + nablaτ(t)22 + Δτ(t)22

Proof By summing side-by-side the first and second in-equalities in Lemma 3 one obtains

ddt

E(t) + c1P(t)le c2 E(t) + E32

(t) + E3(t)1113872 1113873 (65)

Here one has

E32

E34

E34 le

34

E +14E3 (66)

and consequently the right-hand side can be bounded byE + E3 Hence

1113946E(t)

E(0)

dE

E E2 + 1( ) 1113946

E(t)

E(0)

1E

minusE

E2 + 11113874 1113875dEleCt (67)

then

E(t)E2(0) + 1

1113968

E(0)E2(t) + 1

1113968 le eCt

(68)

It follows that once the maximal interval correspondingto a vanishing denominator is identified

forallt isin (0 T)with 2T ≔ logE2(0) + 1

E2(0)1113888 1113889

12C

(69)

E2(t)le

e2CtE2(0)

1 + E2(0) 1 minus e2Ct( )

leE(0)

1 + E2(0)

1113968

1 + E(0) E(0) minus1 + E2(0)

11139681113872 1113873

(70)

8 International Journal of Differential Equations

the energy is boundedNext coming back to (65) and integrating in [0 T) one

deduces a bound for the norm in Linfin(0 T W12(Ω0))capL2(0 T W22(Ω0)) as can be seen hereafter the inequality

1113946T

0P(t)dtleC log

E2(0) + 1E2(0)

1113888 1113889

14

E(0) + E12

(0)1113872 1113873

(71)

implies in particular the integrability of nablav(t)22 and ofΔv(t)22

Although the Galerkin solutions exist for all tgt 0 theiruniform estimates exist only in the finite interval defined in(69) Global-in-time solution will be found only by askingsuitable smallness of Ra and of the initial data

Finally (25) is derived formally with respect to t and thesame procedure is repeated as in the first estimate but for thenonlinear convective terms Now in order to bound theinitial data for vt initial data in W22 is chosen

ΔΠt minus βΠtz minuseβz

Prnabla middot vt middot nablav + ϕtv + v middot nablavt + ϕvt( 1113857

+Raeβzτtz

1Pr

zvt

zt+ vt middot nablav + ϕtv + v middot nablavt + ϕvt1113888 1113889 minus Δvt

minus nabla eminus βzΠt( 1113857 minus βeminus βzΠtk + Raτtk

zτt

zt+ vt middot nablaτ + v middot nablaτt minus Δτt v

zt

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(72)

Again the terms depending on ϕ or ϕt will be boundedexactly as the corresponding convective ones

Lemma 5 Let us define the energy function as

E(t) ≔12

vt

22

Pr+ Ra τt

22

⎛⎝ ⎞⎠ (73)

en for sufficiently small εgt 0 the following inequalityholds truedEdt

(t) + c nablavt

22 + Ra nablaτt

221113874 1113875leC(β PrRa)Δv

22E(t)

+ C(βPr)Raε

vt

22

(74)

where c isin (0 1)

Proof Let us start with the equation

ddt

E(t) + nablavt

22 + Ra nablaτt

22 minus e

minus βznablaΠt vt1113872 1113873 + 2Ra τt vzt( 1113857 +

minus1Pr

vt middot nablav + v middot nablavt vt( 1113857 minus1Pr

ϕtv + ϕvt vt( 1113857 minus Ra vt middot nablaτ + v middot nablaτt τt( 1113857

(75)

Technically the estimates of the nonlinear convectiveterms are the only different ones from the previous onessince the terms are not homogeneous

eminus βznablaΠt vt1113872 1113873

11138681113868111386811138681113868τt0le c vt

2 vt middot nablav

2 + v middot nablavt

21113872 1113873

le c vt

2 vt

3nablav6 + vt

2vinfin nablavt

21113872 1113873

le c vt

322 nablavt

122 Δv2 + vt

2Δv2 nablavt

21113874 1113875

le c Δv22 vt

22 + ε nablavt

221113874 1113875

(76)

e same estimates can be repeated for the other non-linear terms such as

ϕvt vt( 1113857le cnablav6 vt

2 vt

3

ϕtv vt( 1113857le c vt

2vinfin nablavt

2

vt middot nablaτ τt( 1113857le cnablaτ6 vt

2 τt

3

v middot nablaτt τt( 1113857le c τt

2vinfin nablaτt

2

(77)

e gradient terms can be absorbed at the left-handside

Starting from (74) where the positive term at the left-hand side can be neglected and the term vt

22 at the right-

International Journal of Differential Equations 9

hand side can be bounded by E(t) by a generalizedGronwall argument one shows that

E(t)le ec 1113938

T

0Δv(t)22+1( )dt

E(0) (78)

As a matter of fact (71) implies v isin L2(0 T W22(Ω0))so thatE(t) is bounded in (0 T) Further by coming back to(74) and integrating again it follows also

1113946T

0nablavt

22 + nablaτt

221113874 1113875dtleO v0

22 τ0

221113872 1113873 (79)

and this is independent of the size of Ra

Theorem 1 Let βlt 2π and Pr and Ra be arbitrary thensystem (25) with initial data (v0 τ0) isinW22(Ω0) admits asolution a la Ladyzhenskaya namely

(v τ) isin Linfin 0 T W

22 Ω0( 11138571113872 1113873capL2 0 T W

32 Ω0( 11138571113872 1113873

vt τt( 1113857 isin Linfin 0 T L

2 Ω0( 11138571113872 1113873cap L2 0 T W

12 Ω0( 11138571113872 1113873(80)

where

T ≔ logE2(0) + 1

E2(0)1113888 1113889

14C

(81)

Moreover if nabla middot v0 0 then nabla middot v(t) 0 for all t isin (0 T]Finally if Ra v022 and τ022 are sufficiently smalldepending on Pr and β then T infin is taken in (80) and (v τ)

decays exponentially fast to zero

Proof Employing the estimates proved in Lemma 4 it isroutine to show that the approximate solutions constructedwith the Galerkin method converge to a solution

(v τ) isin Linfin 0 T W

12 Ω0( 11138571113872 1113873capL2 0 T W

22 Ω0( 11138571113872 1113873

Π isin L2 0 T W

22 Ω0( 11138571113872 1113873

vt τt( 1113857 isin Linfin 0 T L

2 Ω0( 11138571113872 1113873cap L2 0 T W

12 Ω0( 11138571113872 1113873

(82)

where T is given in (81)e proof of the further regularity properties implying

that the solution fulfils system (17) too is based on (78) and(79) and in particular on the uniform boundedness of vt2as well as the integrability of nablavt(t)22 To this end viewingthe momentum balance as a Poisson problem and increasingthe pressure in the usual way the following is obtained withthe help of Lemma 1

v(t)22 leC v middot nablav2 +nablaΠ2 + vt

2 +τ21113872 1113873

leC nablav42 + εΔv

22 + vt

2 +τ21113872 1113873

(83)

e ε term at the right-hand side can be absorbed by theleft-hand side so that finally the following bound is obtained

ess sup(0T)

v(t)22 leCess sup(0T)

nablav(t)42 +τ2 + vt

21113872 1113873

leO v0

22 τ0

221113872 1113873

(84)

Moreover again from Lemma 1 it is also obtained that

1113946T

0v(t)

232dtleC 1113946

T

0v middot nablav

212 +Π

222 + vt

212 +τ

2121113874 1113875dt

(85)

It is remarked that as noticed in Section 2 the com-patibility condition related to the Neumann problem isverified Here the nonlinear terms at the right-hand side canbe increased by using (14) and interpolation inequality

v middot D2v

2 le cvinfin D

2v

2 le c D2v

22

nablav(nablav)T

le cnablav

24 le cnablav6nablav3 le c D

2v

22

(86)

e proof that τ is bounded in Linfin(0 T W12(Ω0))capL2(0 T W22(Ω0)) is similar and simpler

e proof of global existence follows from Lemmas 3 and5 by setting

E0(t) ≔ E(t) + E(t)

P0(t) ≔ nablav(t)22 +Δv(t)

22 + nablavt(t)

22

+ Ra nablaτ(t)22 +Δτ(t)

22 + nablaτt(t)

221113874 1113875

(87)

In fact if inequalities (49) (51) and (74) are addedbringing to the left-hand side the three terms with factorRaε and using the Poincare inequality

Ele kP (88)

to absorb them in P0 for sufficiently small Ra (dependingon ε) one gets

ddt

E(t) + c1P(t)le c2 E32

(t) + E3(t)1113872 1113873 (89)

e constant c1 is now fixed positive by suitablyrestricting Ra Again using the Poincare inequality on theleft-hand side of (89) one infers

ddt

E(t) + c1 minus c2 E12

(t) + E2(t)1113872 11138731113960 1113961

1k

E(t)le 0 (90)

us the exponential decay follows by choosing at theinitial time

E12

(0) + E2(0)le

12

c1

c2 (91)

In fact by continuity for some time E(t)leE(0) Butthis reinforces the previous condition so that the positivefunction in square brackets cannot decrease and is boundedfrom below by a positive constant for all times

Data Availability

e data that support the findings of this study are availablewithin the article

Conflicts of Interest

e author declares that there are no conflicts of interest

10 International Journal of Differential Equations

References

[1] A Giacobbe and G Mulone ldquoStability in the rotating Benardproblem and its optimal Lyapunov functionsrdquo Acta Appli-candae Mathematicae vol 132 no 1 2014

[2] A Passerini C Ferrario M Ruzicka and G ater ldquoe-oretical results on steady convective flows between horizontalcoaxial cylindersrdquo SIAM Journal on Applied Mathematicsvol 71 no 2 pp 465ndash486 2011

[3] A Passerini and C Ferrario ldquoA theoretical study of the firsttransition for the non-linear Stokes problem in a horizontalannulusrdquo International Journal of Non-linear Mechanicsvol 78 no 1 pp 1ndash8 2015

[4] A De Martino and A Passerini ldquoExistence and nonlinearstability of convective solutions for almost compressible fluidsin Benard problemrdquo Journal of Mathematical Physics vol 60no 11 Article ID 113101 2019

[5] F Crispo and P Maremonti ldquoA high regularity result ofsolutions to modified p-Stokes equationsrdquo Nonlinear Anal-ysis vol 118 pp 97ndash129 2015

[6] H Gouin and T Ruggeri ldquoA consistent thermodynamicalmodel of incompressible media as limit case of quasi-thermal-incompressible materialsrdquo International Journal of Non-Linear Mechanics vol 47 no 6 pp 688ndash693 2012

[7] A Passerini and T Ruggeri ldquoe Benard problem for quasi-thermal-incompressible materials a linear analysisrdquo Inter-national Journal of Non-Linear Mechanics vol 67 no 1pp 178ndash185 2014

[8] S Gatti and E Vuk ldquoSingular limit of equations for linearviscoelastic fluids with periodic boundary conditionsrdquo In-ternational Journal of Non-Linear Mechanics vol 41 no 4pp 518ndash526 2006

[9] A Corli and A Passerini ldquoe Benard problem for slightlycompressible materials existence and linear instabilityrdquoMediterranean Journal of Mathematics vol 16 no 1 2019

[10] G P Galdi ldquoNon-linear stability of the magnetic Benardproblem via a generalized energy methodrdquo Archive for Ra-tional Mechanics and Analysis vol 87 no 2 pp 167ndash1861985

[11] G P Galdi and M Padula ldquoA new approach to energy theoryin the stability of fluid motionrdquo Archive for Rational Me-chanics and Analysis vol 110 no 3 pp 187ndash286 1990

[12] A De Martino and A Passerini ldquoA Lorenz model for almostcompressible fluidsrdquo Mediterranean Journal of Mathematicsvol 17 no 1 p 12 2020

[13] G P Galdi An Introduction to the Mathematical eory of theNavier-Stokes Equation Springer New York NY USA 1998

[14] O Ladyzhenskaya e Mathematical eory of Viscous In-compressible Fluids Gordon amp Breach London UK 1963

[15] G Iooss ldquoeorie non lineaire de la stabilite des ecoulementslaminaires dans le cas de ldquolrsquoechange des stabilitesrdquo Archive forRational Mechanics and Analysis vol 40 no 3 pp 166ndash2081971

[16] M Schechter ldquoCoerciveness in Lprdquo Transactions of theAmerican Mathematical Society vol 107 no 1 pp 10ndash291963

International Journal of Differential Equations 11

After squaring and summing the inequalities derived by(42) and evaluating at the initial time one obtains a boundindependent of N estimated by the W22-norm of the initial

data through Besselrsquos inequality Since the basis is inW22(Ω0) by linearity

ΔvN(0)

2 1113944

N

j1λ(j) v0Ψj1113872 1113873Ψj

2

1113944N

j1v0ΔΨj1113872 1113873Ψj

2

1113944N

j1Δv0Ψj1113872 1113873Ψj

2

le Δv0

2 (45)

Concerning the nonlinear terms one can compute forinstance

1113944

N

j11113944

N

ki11113944

N

ln1Ψk middot nablaΨiΨj1113872 1113873 Ψl middot nablaΨnΨj1113872 1113873C

kN(0)C

iN(0)C

lN(0)C

nN(0)

le 1113944

infin

j1v0 middot nablav0Ψj1113872 1113873

2 v0 middot nablav0

2

(46)

Next these nonlinear terms at the right-hand side can beincreased by (12) and (13) through Holder (p (32) andpprime 3) and interpolation (p (43) and pprime 4) inequalities(the procedure is detailed in the proof of Lemma 3) Finallyusing the Poincare inequality and (10) one gets

vNt (0)

2 leC(β PrRa) Δv0

2 + τ0

2 + Δv0

221113874 1113875

τNt (0)

2 leC(β PrRa) Δτ0

2 + v0

2 + nablav0

2 Δτ0

21113872 1113873

(47)

Here and hereafter the L2-norms of the two nonlinearterms v middot nablav and ϕv are estimated exactly in the same waybut for brevity only the first one will be developed

4 Evolutive Estimates and Main Results

Let us formally derive the a priori estimates for solutions ofsystem (25) by quite a standard procedure they are verifiedjust by Galerkin solutions denoted here by (v τ) and asseen are bound uniformly with respect to N

Lemma 3 Let us define the energy functions

E1(t) ≔12

v22Pr

+ Raτ221113888 1113889

E2(t) ≔12

nablav22Pr

+ Ranablaτ221113888 1113889

(48)

en for sufficiently small εgt 0 the following inequalitieshold true

dE1

dt+ c nablav

22 + Ranablaτ

221113872 1113873leC(β PrRa) E

31 + E

322 + εΔv

221113872 1113873 + C(β Pr)

Raε

v22 (49)

dE2

dt+ c Δv

22 + RaΔτ

221113872 1113873leC(β PrRa)E

32 + C(β Pr)

Raε

E1 (50)

dE2

dt+ c Δv

22 + Raτ

221113872 1113873leC(β PrRa)E

32 + C(β Pr)

Raε

Δv22 (51)

where c isin (0 1) Proof One can test (25) with (v τ) and then one multipliesthe third equation by Ra and then sums

ddt

E1(t) +nablav22 + Ranablaτ

22 minus e

minus βznablaΠ v1113872 1113873 + 2Ra τ vz

( 1113857

minus1Pr

(v middot nablav v) minus1Pr

(ϕv v) minus Ra(v middot nablaτ τ)

(52)

6 International Journal of Differential Equations

In order to estimate the nonlinear term arising from thecoupling with the pressure equation one uses (35) thenonlinear term at the right-hand side can be increased by

(12) and (13) through Holder (p (32) and pprime 3) andinterpolation (p (43) and pprime 4) inequalities so that

eminus βznablaΠ v1113872 1113873

11138681113868111386811138681113868τ0le cv middot nablav2v2 le cv3nablav6v2 le cv

122 v

126 Δv2v2

le cv322 nablav

122 Δv2 le

c

4ε2v

62 +

cε4

nablav22 +

cε2

Δv22

(53)

is term is multiplied by β(C(β)Pr) while the linearpart of the pressure estimate can be included in the estimateof the buoyancy term and of the energy equation transportterm giving rise (in the whole) to

RaC(β Pr)τ2v2 leRaC(β Pr)ε2τ

22 +

12ε

v221113874 1113875 (54)

Here the termwith the second derivatives does not allowto get an autonomous bound for E1(t) so that a furtherestimate for higher derivatives need to be added Notice thatthe convective terms do not vanish and by also resorting toYoungrsquos inequality they are bounded as follows

(v middot nablav v) 1113946Ω0nabla middot v

v2

21113888 1113889 minus 1113946

Ω0ϕ

v2

2le12ϕ2v

24

leCnablav2v122 v

326 leCnablav

32

(v middot nablaτ τ) 1113946Ω0nabla middot v

τ2

21113888 1113889 minus 1113946

Ω0ϕτ2

2le12ϕ2τ

24

leCnablav2τ122 τ

326 leCnablav2nablaτ

22 le

13

Cnablav32 +

23

Cnablaτ32

(55)

Analogously

1113946Ω0ϕv

2 le nablav32 (56)

By collecting all these estimates using the Poincareinequality increasing nablav32 with E32

2 and absorbing at the

left-hand side where possible the terms of order ϵ the firstinequality follows

Now the balance equations are tested with Δv and Δτrespectively and again (35) is inserted

First the following equations are written

12Pr

ddt

nablav22 +Δv

22

1Pr

(v middot nablav + ϕvΔv) + eminus βznablaΠΔv1113872 1113873 minus Ra(τkΔv)

12ddt

nablaτ22 +Δτ

22 (v middot nablaτΔτ) + v

zΔτ( 1113857

dE2

dt+Δv

22 + RaΔτ

22 e

minus βznablaΠΔv1113872 1113873 + Ra τΔvz( 1113857 + Ra Δτ v

z( 1113857

1Pr

(v middot nablavΔv) +1Pr

(ϕvΔv) + Ra(v middot nablaτΔτ)

(57)

Again the pressure-related convective terms arebounded and then the temperature-depending couplingterms are estimated

eminus βznablaΠΔv1113872 1113873

11138681113868111386811138681113868τ0le cv middot nablav2Δv2 le v6nablav3Δv2 le

cv6nablav122 D

2v

122 Δv2 le cnablav

322 Δv

322 le

3cε4

Δv22 +

c

4εnablav

62

(58)

International Journal of Differential Equations 7

In order to reach local-in-time existence of solutions forany initial data one writes

RaC(β Pr) τ2Δv2 +Δτ2v2( 1113857leRaC(β Pr)ε2Δτ

22 +

ε2Δv

22 +

12ε2

E1(t)1113874 1113875 (59)

In order to achieve global-in-time existence for smallinitial data one uses an alternative estimate which followsby several applications of the Poincare inequality

RaC(β Pr) τ2Δv2 +Δτ2v2( 1113857

leRaC(β Pr)ε2Δτ

22 +

12ε

Δv221113874 1113875

(60)

Now the nonlinear terms are focused

(v middot nablav + ϕvΔv)leCv6nablav3Δv2 leCv6nablav122 nablav

126 Δv2

leCnablav322 Δv

322 le

3ε4

CΔv22 +

14ε

Cnablav62

(v middot nablaτΔτ)le v6nablaτ3Δτ2 leCnablav2nablaτ122 D

2τ

122 Δτ2

leCnablav2nablaτ122 Δτ

322 le

3ε4

CΔτ22 +

14ε

Cnablaτ22nablav

42

le3ε4

CΔτ22 +

112ε

Cnablaτ62 +

16ε

Cnablav62

(61)

e terms of order ϵ are still absorbed at the left-handside and then inequalities (50) and (51) respectively followby (59) and (60)

Lemma 4 Let

T ≔ logE2(0) + 1

E2(0)1113888 1113889

14C

(62)

then for all t isin (0 T] the following estimates hold

E(t)le E(0)

1 + E2(0)

1113968

1 + E(0) E(0) minus1 + E2(0)

11139681113872 1113873

⎛⎝ ⎞⎠

12

(63)

where E(t) ≔ E1(t) + E2(t) Furthermore

1113946T

0P(t)dtleC log

E2(0) + 1E2(0)

1113888 1113889

14

E(0) + E12

(0)1113872 1113873

(64)

where P(t) ≔ nablav(t)22 + Δv(t)22 + nablaτ(t)22 + Δτ(t)22

Proof By summing side-by-side the first and second in-equalities in Lemma 3 one obtains

ddt

E(t) + c1P(t)le c2 E(t) + E32

(t) + E3(t)1113872 1113873 (65)

Here one has

E32

E34

E34 le

34

E +14E3 (66)

and consequently the right-hand side can be bounded byE + E3 Hence

1113946E(t)

E(0)

dE

E E2 + 1( ) 1113946

E(t)

E(0)

1E

minusE

E2 + 11113874 1113875dEleCt (67)

then

E(t)E2(0) + 1

1113968

E(0)E2(t) + 1

1113968 le eCt

(68)

It follows that once the maximal interval correspondingto a vanishing denominator is identified

forallt isin (0 T)with 2T ≔ logE2(0) + 1

E2(0)1113888 1113889

12C

(69)

E2(t)le

e2CtE2(0)

1 + E2(0) 1 minus e2Ct( )

leE(0)

1 + E2(0)

1113968

1 + E(0) E(0) minus1 + E2(0)

11139681113872 1113873

(70)

8 International Journal of Differential Equations

the energy is boundedNext coming back to (65) and integrating in [0 T) one

deduces a bound for the norm in Linfin(0 T W12(Ω0))capL2(0 T W22(Ω0)) as can be seen hereafter the inequality

1113946T

0P(t)dtleC log

E2(0) + 1E2(0)

1113888 1113889

14

E(0) + E12

(0)1113872 1113873

(71)

implies in particular the integrability of nablav(t)22 and ofΔv(t)22

Although the Galerkin solutions exist for all tgt 0 theiruniform estimates exist only in the finite interval defined in(69) Global-in-time solution will be found only by askingsuitable smallness of Ra and of the initial data

Finally (25) is derived formally with respect to t and thesame procedure is repeated as in the first estimate but for thenonlinear convective terms Now in order to bound theinitial data for vt initial data in W22 is chosen

ΔΠt minus βΠtz minuseβz

Prnabla middot vt middot nablav + ϕtv + v middot nablavt + ϕvt( 1113857

+Raeβzτtz

1Pr

zvt

zt+ vt middot nablav + ϕtv + v middot nablavt + ϕvt1113888 1113889 minus Δvt

minus nabla eminus βzΠt( 1113857 minus βeminus βzΠtk + Raτtk

zτt

zt+ vt middot nablaτ + v middot nablaτt minus Δτt v

zt

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(72)

Again the terms depending on ϕ or ϕt will be boundedexactly as the corresponding convective ones

Lemma 5 Let us define the energy function as

E(t) ≔12

vt

22

Pr+ Ra τt

22

⎛⎝ ⎞⎠ (73)

en for sufficiently small εgt 0 the following inequalityholds truedEdt

(t) + c nablavt

22 + Ra nablaτt

221113874 1113875leC(β PrRa)Δv

22E(t)

+ C(βPr)Raε

vt

22

(74)

where c isin (0 1)

Proof Let us start with the equation

ddt

E(t) + nablavt

22 + Ra nablaτt

22 minus e

minus βznablaΠt vt1113872 1113873 + 2Ra τt vzt( 1113857 +

minus1Pr

vt middot nablav + v middot nablavt vt( 1113857 minus1Pr

ϕtv + ϕvt vt( 1113857 minus Ra vt middot nablaτ + v middot nablaτt τt( 1113857

(75)

Technically the estimates of the nonlinear convectiveterms are the only different ones from the previous onessince the terms are not homogeneous

eminus βznablaΠt vt1113872 1113873

11138681113868111386811138681113868τt0le c vt

2 vt middot nablav

2 + v middot nablavt

21113872 1113873

le c vt

2 vt

3nablav6 + vt

2vinfin nablavt

21113872 1113873

le c vt

322 nablavt

122 Δv2 + vt

2Δv2 nablavt

21113874 1113875

le c Δv22 vt

22 + ε nablavt

221113874 1113875

(76)

e same estimates can be repeated for the other non-linear terms such as

ϕvt vt( 1113857le cnablav6 vt

2 vt

3

ϕtv vt( 1113857le c vt

2vinfin nablavt

2

vt middot nablaτ τt( 1113857le cnablaτ6 vt

2 τt

3

v middot nablaτt τt( 1113857le c τt

2vinfin nablaτt

2

(77)

e gradient terms can be absorbed at the left-handside

Starting from (74) where the positive term at the left-hand side can be neglected and the term vt

22 at the right-

International Journal of Differential Equations 9

hand side can be bounded by E(t) by a generalizedGronwall argument one shows that

E(t)le ec 1113938

T

0Δv(t)22+1( )dt

E(0) (78)

As a matter of fact (71) implies v isin L2(0 T W22(Ω0))so thatE(t) is bounded in (0 T) Further by coming back to(74) and integrating again it follows also

1113946T

0nablavt

22 + nablaτt

221113874 1113875dtleO v0

22 τ0

221113872 1113873 (79)

and this is independent of the size of Ra

Theorem 1 Let βlt 2π and Pr and Ra be arbitrary thensystem (25) with initial data (v0 τ0) isinW22(Ω0) admits asolution a la Ladyzhenskaya namely

(v τ) isin Linfin 0 T W

22 Ω0( 11138571113872 1113873capL2 0 T W

32 Ω0( 11138571113872 1113873

vt τt( 1113857 isin Linfin 0 T L

2 Ω0( 11138571113872 1113873cap L2 0 T W

12 Ω0( 11138571113872 1113873(80)

where

T ≔ logE2(0) + 1

E2(0)1113888 1113889

14C

(81)

Moreover if nabla middot v0 0 then nabla middot v(t) 0 for all t isin (0 T]Finally if Ra v022 and τ022 are sufficiently smalldepending on Pr and β then T infin is taken in (80) and (v τ)

decays exponentially fast to zero

Proof Employing the estimates proved in Lemma 4 it isroutine to show that the approximate solutions constructedwith the Galerkin method converge to a solution

(v τ) isin Linfin 0 T W

12 Ω0( 11138571113872 1113873capL2 0 T W

22 Ω0( 11138571113872 1113873

Π isin L2 0 T W

22 Ω0( 11138571113872 1113873

vt τt( 1113857 isin Linfin 0 T L

2 Ω0( 11138571113872 1113873cap L2 0 T W

12 Ω0( 11138571113872 1113873

(82)

where T is given in (81)e proof of the further regularity properties implying

that the solution fulfils system (17) too is based on (78) and(79) and in particular on the uniform boundedness of vt2as well as the integrability of nablavt(t)22 To this end viewingthe momentum balance as a Poisson problem and increasingthe pressure in the usual way the following is obtained withthe help of Lemma 1

v(t)22 leC v middot nablav2 +nablaΠ2 + vt

2 +τ21113872 1113873

leC nablav42 + εΔv

22 + vt

2 +τ21113872 1113873

(83)

e ε term at the right-hand side can be absorbed by theleft-hand side so that finally the following bound is obtained

ess sup(0T)

v(t)22 leCess sup(0T)

nablav(t)42 +τ2 + vt

21113872 1113873

leO v0

22 τ0

221113872 1113873

(84)

Moreover again from Lemma 1 it is also obtained that

1113946T

0v(t)

232dtleC 1113946

T

0v middot nablav

212 +Π

222 + vt

212 +τ

2121113874 1113875dt

(85)

It is remarked that as noticed in Section 2 the com-patibility condition related to the Neumann problem isverified Here the nonlinear terms at the right-hand side canbe increased by using (14) and interpolation inequality

v middot D2v

2 le cvinfin D

2v

2 le c D2v

22

nablav(nablav)T

le cnablav

24 le cnablav6nablav3 le c D

2v

22

(86)

e proof that τ is bounded in Linfin(0 T W12(Ω0))capL2(0 T W22(Ω0)) is similar and simpler

e proof of global existence follows from Lemmas 3 and5 by setting

E0(t) ≔ E(t) + E(t)

P0(t) ≔ nablav(t)22 +Δv(t)

22 + nablavt(t)

22

+ Ra nablaτ(t)22 +Δτ(t)

22 + nablaτt(t)

221113874 1113875

(87)

In fact if inequalities (49) (51) and (74) are addedbringing to the left-hand side the three terms with factorRaε and using the Poincare inequality

Ele kP (88)

to absorb them in P0 for sufficiently small Ra (dependingon ε) one gets

ddt

E(t) + c1P(t)le c2 E32

(t) + E3(t)1113872 1113873 (89)

e constant c1 is now fixed positive by suitablyrestricting Ra Again using the Poincare inequality on theleft-hand side of (89) one infers

ddt

E(t) + c1 minus c2 E12

(t) + E2(t)1113872 11138731113960 1113961

1k

E(t)le 0 (90)

us the exponential decay follows by choosing at theinitial time

E12

(0) + E2(0)le

12

c1

c2 (91)

In fact by continuity for some time E(t)leE(0) Butthis reinforces the previous condition so that the positivefunction in square brackets cannot decrease and is boundedfrom below by a positive constant for all times

Data Availability

e data that support the findings of this study are availablewithin the article

Conflicts of Interest

e author declares that there are no conflicts of interest

10 International Journal of Differential Equations

References

[1] A Giacobbe and G Mulone ldquoStability in the rotating Benardproblem and its optimal Lyapunov functionsrdquo Acta Appli-candae Mathematicae vol 132 no 1 2014

[2] A Passerini C Ferrario M Ruzicka and G ater ldquoe-oretical results on steady convective flows between horizontalcoaxial cylindersrdquo SIAM Journal on Applied Mathematicsvol 71 no 2 pp 465ndash486 2011

[3] A Passerini and C Ferrario ldquoA theoretical study of the firsttransition for the non-linear Stokes problem in a horizontalannulusrdquo International Journal of Non-linear Mechanicsvol 78 no 1 pp 1ndash8 2015

[4] A De Martino and A Passerini ldquoExistence and nonlinearstability of convective solutions for almost compressible fluidsin Benard problemrdquo Journal of Mathematical Physics vol 60no 11 Article ID 113101 2019

[5] F Crispo and P Maremonti ldquoA high regularity result ofsolutions to modified p-Stokes equationsrdquo Nonlinear Anal-ysis vol 118 pp 97ndash129 2015

[6] H Gouin and T Ruggeri ldquoA consistent thermodynamicalmodel of incompressible media as limit case of quasi-thermal-incompressible materialsrdquo International Journal of Non-Linear Mechanics vol 47 no 6 pp 688ndash693 2012

[7] A Passerini and T Ruggeri ldquoe Benard problem for quasi-thermal-incompressible materials a linear analysisrdquo Inter-national Journal of Non-Linear Mechanics vol 67 no 1pp 178ndash185 2014

[8] S Gatti and E Vuk ldquoSingular limit of equations for linearviscoelastic fluids with periodic boundary conditionsrdquo In-ternational Journal of Non-Linear Mechanics vol 41 no 4pp 518ndash526 2006

[9] A Corli and A Passerini ldquoe Benard problem for slightlycompressible materials existence and linear instabilityrdquoMediterranean Journal of Mathematics vol 16 no 1 2019

[10] G P Galdi ldquoNon-linear stability of the magnetic Benardproblem via a generalized energy methodrdquo Archive for Ra-tional Mechanics and Analysis vol 87 no 2 pp 167ndash1861985

[11] G P Galdi and M Padula ldquoA new approach to energy theoryin the stability of fluid motionrdquo Archive for Rational Me-chanics and Analysis vol 110 no 3 pp 187ndash286 1990

[12] A De Martino and A Passerini ldquoA Lorenz model for almostcompressible fluidsrdquo Mediterranean Journal of Mathematicsvol 17 no 1 p 12 2020

[13] G P Galdi An Introduction to the Mathematical eory of theNavier-Stokes Equation Springer New York NY USA 1998

[14] O Ladyzhenskaya e Mathematical eory of Viscous In-compressible Fluids Gordon amp Breach London UK 1963

[15] G Iooss ldquoeorie non lineaire de la stabilite des ecoulementslaminaires dans le cas de ldquolrsquoechange des stabilitesrdquo Archive forRational Mechanics and Analysis vol 40 no 3 pp 166ndash2081971

[16] M Schechter ldquoCoerciveness in Lprdquo Transactions of theAmerican Mathematical Society vol 107 no 1 pp 10ndash291963

International Journal of Differential Equations 11

In order to estimate the nonlinear term arising from thecoupling with the pressure equation one uses (35) thenonlinear term at the right-hand side can be increased by

(12) and (13) through Holder (p (32) and pprime 3) andinterpolation (p (43) and pprime 4) inequalities so that

eminus βznablaΠ v1113872 1113873

11138681113868111386811138681113868τ0le cv middot nablav2v2 le cv3nablav6v2 le cv

122 v

126 Δv2v2

le cv322 nablav

122 Δv2 le

c

4ε2v

62 +

cε4

nablav22 +

cε2

Δv22

(53)

is term is multiplied by β(C(β)Pr) while the linearpart of the pressure estimate can be included in the estimateof the buoyancy term and of the energy equation transportterm giving rise (in the whole) to

RaC(β Pr)τ2v2 leRaC(β Pr)ε2τ

22 +

12ε

v221113874 1113875 (54)

Here the termwith the second derivatives does not allowto get an autonomous bound for E1(t) so that a furtherestimate for higher derivatives need to be added Notice thatthe convective terms do not vanish and by also resorting toYoungrsquos inequality they are bounded as follows

(v middot nablav v) 1113946Ω0nabla middot v

v2

21113888 1113889 minus 1113946

Ω0ϕ

v2

2le12ϕ2v

24

leCnablav2v122 v

326 leCnablav

32

(v middot nablaτ τ) 1113946Ω0nabla middot v

τ2

21113888 1113889 minus 1113946

Ω0ϕτ2

2le12ϕ2τ

24

leCnablav2τ122 τ

326 leCnablav2nablaτ

22 le

13

Cnablav32 +

23

Cnablaτ32

(55)

Analogously

1113946Ω0ϕv

2 le nablav32 (56)

By collecting all these estimates using the Poincareinequality increasing nablav32 with E32

2 and absorbing at the

left-hand side where possible the terms of order ϵ the firstinequality follows

Now the balance equations are tested with Δv and Δτrespectively and again (35) is inserted

First the following equations are written

12Pr

ddt

nablav22 +Δv

22

1Pr

(v middot nablav + ϕvΔv) + eminus βznablaΠΔv1113872 1113873 minus Ra(τkΔv)

12ddt

nablaτ22 +Δτ

22 (v middot nablaτΔτ) + v

zΔτ( 1113857

dE2

dt+Δv

22 + RaΔτ

22 e

minus βznablaΠΔv1113872 1113873 + Ra τΔvz( 1113857 + Ra Δτ v

z( 1113857

1Pr

(v middot nablavΔv) +1Pr

(ϕvΔv) + Ra(v middot nablaτΔτ)

(57)

Again the pressure-related convective terms arebounded and then the temperature-depending couplingterms are estimated

eminus βznablaΠΔv1113872 1113873

11138681113868111386811138681113868τ0le cv middot nablav2Δv2 le v6nablav3Δv2 le

cv6nablav122 D

2v

122 Δv2 le cnablav

322 Δv

322 le

3cε4

Δv22 +

c

4εnablav

62

(58)

International Journal of Differential Equations 7

In order to reach local-in-time existence of solutions forany initial data one writes

RaC(β Pr) τ2Δv2 +Δτ2v2( 1113857leRaC(β Pr)ε2Δτ

22 +

ε2Δv

22 +

12ε2

E1(t)1113874 1113875 (59)

In order to achieve global-in-time existence for smallinitial data one uses an alternative estimate which followsby several applications of the Poincare inequality

RaC(β Pr) τ2Δv2 +Δτ2v2( 1113857

leRaC(β Pr)ε2Δτ

22 +

12ε

Δv221113874 1113875

(60)

Now the nonlinear terms are focused

(v middot nablav + ϕvΔv)leCv6nablav3Δv2 leCv6nablav122 nablav

126 Δv2

leCnablav322 Δv

322 le

3ε4

CΔv22 +

14ε

Cnablav62

(v middot nablaτΔτ)le v6nablaτ3Δτ2 leCnablav2nablaτ122 D

2τ

122 Δτ2

leCnablav2nablaτ122 Δτ

322 le

3ε4

CΔτ22 +

14ε

Cnablaτ22nablav

42

le3ε4

CΔτ22 +

112ε

Cnablaτ62 +

16ε

Cnablav62

(61)

e terms of order ϵ are still absorbed at the left-handside and then inequalities (50) and (51) respectively followby (59) and (60)

Lemma 4 Let

T ≔ logE2(0) + 1

E2(0)1113888 1113889

14C

(62)

then for all t isin (0 T] the following estimates hold

E(t)le E(0)

1 + E2(0)

1113968

1 + E(0) E(0) minus1 + E2(0)

11139681113872 1113873

⎛⎝ ⎞⎠

12

(63)

where E(t) ≔ E1(t) + E2(t) Furthermore

1113946T

0P(t)dtleC log

E2(0) + 1E2(0)

1113888 1113889

14

E(0) + E12

(0)1113872 1113873

(64)

where P(t) ≔ nablav(t)22 + Δv(t)22 + nablaτ(t)22 + Δτ(t)22

Proof By summing side-by-side the first and second in-equalities in Lemma 3 one obtains

ddt

E(t) + c1P(t)le c2 E(t) + E32

(t) + E3(t)1113872 1113873 (65)

Here one has

E32

E34

E34 le

34

E +14E3 (66)

and consequently the right-hand side can be bounded byE + E3 Hence

1113946E(t)

E(0)

dE

E E2 + 1( ) 1113946

E(t)

E(0)

1E

minusE

E2 + 11113874 1113875dEleCt (67)

then

E(t)E2(0) + 1

1113968

E(0)E2(t) + 1

1113968 le eCt

(68)

It follows that once the maximal interval correspondingto a vanishing denominator is identified

forallt isin (0 T)with 2T ≔ logE2(0) + 1

E2(0)1113888 1113889

12C

(69)

E2(t)le

e2CtE2(0)

1 + E2(0) 1 minus e2Ct( )

leE(0)

1 + E2(0)

1113968

1 + E(0) E(0) minus1 + E2(0)

11139681113872 1113873

(70)

8 International Journal of Differential Equations

the energy is boundedNext coming back to (65) and integrating in [0 T) one

deduces a bound for the norm in Linfin(0 T W12(Ω0))capL2(0 T W22(Ω0)) as can be seen hereafter the inequality

1113946T

0P(t)dtleC log

E2(0) + 1E2(0)

1113888 1113889

14

E(0) + E12

(0)1113872 1113873

(71)

implies in particular the integrability of nablav(t)22 and ofΔv(t)22

Although the Galerkin solutions exist for all tgt 0 theiruniform estimates exist only in the finite interval defined in(69) Global-in-time solution will be found only by askingsuitable smallness of Ra and of the initial data

Finally (25) is derived formally with respect to t and thesame procedure is repeated as in the first estimate but for thenonlinear convective terms Now in order to bound theinitial data for vt initial data in W22 is chosen

ΔΠt minus βΠtz minuseβz

Prnabla middot vt middot nablav + ϕtv + v middot nablavt + ϕvt( 1113857

+Raeβzτtz

1Pr

zvt

zt+ vt middot nablav + ϕtv + v middot nablavt + ϕvt1113888 1113889 minus Δvt

minus nabla eminus βzΠt( 1113857 minus βeminus βzΠtk + Raτtk

zτt

zt+ vt middot nablaτ + v middot nablaτt minus Δτt v

zt

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(72)

Again the terms depending on ϕ or ϕt will be boundedexactly as the corresponding convective ones

Lemma 5 Let us define the energy function as

E(t) ≔12

vt

22

Pr+ Ra τt

22

⎛⎝ ⎞⎠ (73)

en for sufficiently small εgt 0 the following inequalityholds truedEdt

(t) + c nablavt

22 + Ra nablaτt

221113874 1113875leC(β PrRa)Δv

22E(t)

+ C(βPr)Raε

vt

22

(74)

where c isin (0 1)

Proof Let us start with the equation

ddt

E(t) + nablavt

22 + Ra nablaτt

22 minus e

minus βznablaΠt vt1113872 1113873 + 2Ra τt vzt( 1113857 +

minus1Pr

vt middot nablav + v middot nablavt vt( 1113857 minus1Pr

ϕtv + ϕvt vt( 1113857 minus Ra vt middot nablaτ + v middot nablaτt τt( 1113857

(75)

Technically the estimates of the nonlinear convectiveterms are the only different ones from the previous onessince the terms are not homogeneous

eminus βznablaΠt vt1113872 1113873

11138681113868111386811138681113868τt0le c vt

2 vt middot nablav

2 + v middot nablavt

21113872 1113873

le c vt

2 vt

3nablav6 + vt

2vinfin nablavt

21113872 1113873

le c vt

322 nablavt

122 Δv2 + vt

2Δv2 nablavt

21113874 1113875

le c Δv22 vt

22 + ε nablavt

221113874 1113875

(76)

e same estimates can be repeated for the other non-linear terms such as

ϕvt vt( 1113857le cnablav6 vt

2 vt

3

ϕtv vt( 1113857le c vt

2vinfin nablavt

2

vt middot nablaτ τt( 1113857le cnablaτ6 vt

2 τt

3

v middot nablaτt τt( 1113857le c τt

2vinfin nablaτt

2

(77)

e gradient terms can be absorbed at the left-handside

Starting from (74) where the positive term at the left-hand side can be neglected and the term vt

22 at the right-

International Journal of Differential Equations 9

hand side can be bounded by E(t) by a generalizedGronwall argument one shows that

E(t)le ec 1113938

T

0Δv(t)22+1( )dt

E(0) (78)

As a matter of fact (71) implies v isin L2(0 T W22(Ω0))so thatE(t) is bounded in (0 T) Further by coming back to(74) and integrating again it follows also

1113946T

0nablavt

22 + nablaτt

221113874 1113875dtleO v0

22 τ0

221113872 1113873 (79)

and this is independent of the size of Ra

Theorem 1 Let βlt 2π and Pr and Ra be arbitrary thensystem (25) with initial data (v0 τ0) isinW22(Ω0) admits asolution a la Ladyzhenskaya namely

(v τ) isin Linfin 0 T W

22 Ω0( 11138571113872 1113873capL2 0 T W

32 Ω0( 11138571113872 1113873

vt τt( 1113857 isin Linfin 0 T L

2 Ω0( 11138571113872 1113873cap L2 0 T W

12 Ω0( 11138571113872 1113873(80)

where

T ≔ logE2(0) + 1

E2(0)1113888 1113889

14C

(81)

Moreover if nabla middot v0 0 then nabla middot v(t) 0 for all t isin (0 T]Finally if Ra v022 and τ022 are sufficiently smalldepending on Pr and β then T infin is taken in (80) and (v τ)

decays exponentially fast to zero

Proof Employing the estimates proved in Lemma 4 it isroutine to show that the approximate solutions constructedwith the Galerkin method converge to a solution

(v τ) isin Linfin 0 T W

12 Ω0( 11138571113872 1113873capL2 0 T W

22 Ω0( 11138571113872 1113873

Π isin L2 0 T W

22 Ω0( 11138571113872 1113873

vt τt( 1113857 isin Linfin 0 T L

2 Ω0( 11138571113872 1113873cap L2 0 T W

12 Ω0( 11138571113872 1113873

(82)

where T is given in (81)e proof of the further regularity properties implying

that the solution fulfils system (17) too is based on (78) and(79) and in particular on the uniform boundedness of vt2as well as the integrability of nablavt(t)22 To this end viewingthe momentum balance as a Poisson problem and increasingthe pressure in the usual way the following is obtained withthe help of Lemma 1

v(t)22 leC v middot nablav2 +nablaΠ2 + vt

2 +τ21113872 1113873

leC nablav42 + εΔv

22 + vt

2 +τ21113872 1113873

(83)

e ε term at the right-hand side can be absorbed by theleft-hand side so that finally the following bound is obtained

ess sup(0T)

v(t)22 leCess sup(0T)

nablav(t)42 +τ2 + vt

21113872 1113873

leO v0

22 τ0

221113872 1113873

(84)

Moreover again from Lemma 1 it is also obtained that

1113946T

0v(t)

232dtleC 1113946

T

0v middot nablav

212 +Π

222 + vt

212 +τ

2121113874 1113875dt

(85)

It is remarked that as noticed in Section 2 the com-patibility condition related to the Neumann problem isverified Here the nonlinear terms at the right-hand side canbe increased by using (14) and interpolation inequality

v middot D2v

2 le cvinfin D

2v

2 le c D2v

22

nablav(nablav)T

le cnablav

24 le cnablav6nablav3 le c D

2v

22

(86)

e proof that τ is bounded in Linfin(0 T W12(Ω0))capL2(0 T W22(Ω0)) is similar and simpler

e proof of global existence follows from Lemmas 3 and5 by setting

E0(t) ≔ E(t) + E(t)

P0(t) ≔ nablav(t)22 +Δv(t)

22 + nablavt(t)

22

+ Ra nablaτ(t)22 +Δτ(t)

22 + nablaτt(t)

221113874 1113875

(87)

In fact if inequalities (49) (51) and (74) are addedbringing to the left-hand side the three terms with factorRaε and using the Poincare inequality

Ele kP (88)

to absorb them in P0 for sufficiently small Ra (dependingon ε) one gets

ddt

E(t) + c1P(t)le c2 E32

(t) + E3(t)1113872 1113873 (89)

e constant c1 is now fixed positive by suitablyrestricting Ra Again using the Poincare inequality on theleft-hand side of (89) one infers

ddt

E(t) + c1 minus c2 E12

(t) + E2(t)1113872 11138731113960 1113961

1k

E(t)le 0 (90)

us the exponential decay follows by choosing at theinitial time

E12

(0) + E2(0)le

12

c1

c2 (91)

In fact by continuity for some time E(t)leE(0) Butthis reinforces the previous condition so that the positivefunction in square brackets cannot decrease and is boundedfrom below by a positive constant for all times

Data Availability

e data that support the findings of this study are availablewithin the article

Conflicts of Interest

e author declares that there are no conflicts of interest

10 International Journal of Differential Equations

References

[1] A Giacobbe and G Mulone ldquoStability in the rotating Benardproblem and its optimal Lyapunov functionsrdquo Acta Appli-candae Mathematicae vol 132 no 1 2014

[2] A Passerini C Ferrario M Ruzicka and G ater ldquoe-oretical results on steady convective flows between horizontalcoaxial cylindersrdquo SIAM Journal on Applied Mathematicsvol 71 no 2 pp 465ndash486 2011

[3] A Passerini and C Ferrario ldquoA theoretical study of the firsttransition for the non-linear Stokes problem in a horizontalannulusrdquo International Journal of Non-linear Mechanicsvol 78 no 1 pp 1ndash8 2015

[4] A De Martino and A Passerini ldquoExistence and nonlinearstability of convective solutions for almost compressible fluidsin Benard problemrdquo Journal of Mathematical Physics vol 60no 11 Article ID 113101 2019

[5] F Crispo and P Maremonti ldquoA high regularity result ofsolutions to modified p-Stokes equationsrdquo Nonlinear Anal-ysis vol 118 pp 97ndash129 2015

[6] H Gouin and T Ruggeri ldquoA consistent thermodynamicalmodel of incompressible media as limit case of quasi-thermal-incompressible materialsrdquo International Journal of Non-Linear Mechanics vol 47 no 6 pp 688ndash693 2012

[7] A Passerini and T Ruggeri ldquoe Benard problem for quasi-thermal-incompressible materials a linear analysisrdquo Inter-national Journal of Non-Linear Mechanics vol 67 no 1pp 178ndash185 2014

[8] S Gatti and E Vuk ldquoSingular limit of equations for linearviscoelastic fluids with periodic boundary conditionsrdquo In-ternational Journal of Non-Linear Mechanics vol 41 no 4pp 518ndash526 2006

[9] A Corli and A Passerini ldquoe Benard problem for slightlycompressible materials existence and linear instabilityrdquoMediterranean Journal of Mathematics vol 16 no 1 2019

[10] G P Galdi ldquoNon-linear stability of the magnetic Benardproblem via a generalized energy methodrdquo Archive for Ra-tional Mechanics and Analysis vol 87 no 2 pp 167ndash1861985

[11] G P Galdi and M Padula ldquoA new approach to energy theoryin the stability of fluid motionrdquo Archive for Rational Me-chanics and Analysis vol 110 no 3 pp 187ndash286 1990

[12] A De Martino and A Passerini ldquoA Lorenz model for almostcompressible fluidsrdquo Mediterranean Journal of Mathematicsvol 17 no 1 p 12 2020

[13] G P Galdi An Introduction to the Mathematical eory of theNavier-Stokes Equation Springer New York NY USA 1998

[14] O Ladyzhenskaya e Mathematical eory of Viscous In-compressible Fluids Gordon amp Breach London UK 1963

[15] G Iooss ldquoeorie non lineaire de la stabilite des ecoulementslaminaires dans le cas de ldquolrsquoechange des stabilitesrdquo Archive forRational Mechanics and Analysis vol 40 no 3 pp 166ndash2081971

[16] M Schechter ldquoCoerciveness in Lprdquo Transactions of theAmerican Mathematical Society vol 107 no 1 pp 10ndash291963

International Journal of Differential Equations 11

In order to reach local-in-time existence of solutions forany initial data one writes

RaC(β Pr) τ2Δv2 +Δτ2v2( 1113857leRaC(β Pr)ε2Δτ

22 +

ε2Δv

22 +

12ε2

E1(t)1113874 1113875 (59)

In order to achieve global-in-time existence for smallinitial data one uses an alternative estimate which followsby several applications of the Poincare inequality

RaC(β Pr) τ2Δv2 +Δτ2v2( 1113857

leRaC(β Pr)ε2Δτ

22 +

12ε

Δv221113874 1113875

(60)

Now the nonlinear terms are focused

(v middot nablav + ϕvΔv)leCv6nablav3Δv2 leCv6nablav122 nablav

126 Δv2

leCnablav322 Δv

322 le

3ε4

CΔv22 +

14ε

Cnablav62

(v middot nablaτΔτ)le v6nablaτ3Δτ2 leCnablav2nablaτ122 D

2τ

122 Δτ2

leCnablav2nablaτ122 Δτ

322 le

3ε4

CΔτ22 +

14ε

Cnablaτ22nablav

42

le3ε4

CΔτ22 +

112ε

Cnablaτ62 +

16ε

Cnablav62

(61)

e terms of order ϵ are still absorbed at the left-handside and then inequalities (50) and (51) respectively followby (59) and (60)

Lemma 4 Let

T ≔ logE2(0) + 1

E2(0)1113888 1113889

14C

(62)

then for all t isin (0 T] the following estimates hold

E(t)le E(0)

1 + E2(0)

1113968

1 + E(0) E(0) minus1 + E2(0)

11139681113872 1113873

⎛⎝ ⎞⎠

12

(63)

where E(t) ≔ E1(t) + E2(t) Furthermore

1113946T

0P(t)dtleC log

E2(0) + 1E2(0)

1113888 1113889

14

E(0) + E12

(0)1113872 1113873

(64)

where P(t) ≔ nablav(t)22 + Δv(t)22 + nablaτ(t)22 + Δτ(t)22

Proof By summing side-by-side the first and second in-equalities in Lemma 3 one obtains

ddt

E(t) + c1P(t)le c2 E(t) + E32

(t) + E3(t)1113872 1113873 (65)

Here one has

E32

E34

E34 le

34

E +14E3 (66)

and consequently the right-hand side can be bounded byE + E3 Hence

1113946E(t)

E(0)

dE

E E2 + 1( ) 1113946

E(t)

E(0)

1E

minusE

E2 + 11113874 1113875dEleCt (67)

then

E(t)E2(0) + 1

1113968

E(0)E2(t) + 1

1113968 le eCt

(68)

It follows that once the maximal interval correspondingto a vanishing denominator is identified

forallt isin (0 T)with 2T ≔ logE2(0) + 1

E2(0)1113888 1113889

12C

(69)

E2(t)le

e2CtE2(0)

1 + E2(0) 1 minus e2Ct( )

leE(0)

1 + E2(0)

1113968

1 + E(0) E(0) minus1 + E2(0)

11139681113872 1113873

(70)

8 International Journal of Differential Equations

the energy is boundedNext coming back to (65) and integrating in [0 T) one

deduces a bound for the norm in Linfin(0 T W12(Ω0))capL2(0 T W22(Ω0)) as can be seen hereafter the inequality

1113946T

0P(t)dtleC log

E2(0) + 1E2(0)

1113888 1113889

14

E(0) + E12

(0)1113872 1113873

(71)

implies in particular the integrability of nablav(t)22 and ofΔv(t)22

Although the Galerkin solutions exist for all tgt 0 theiruniform estimates exist only in the finite interval defined in(69) Global-in-time solution will be found only by askingsuitable smallness of Ra and of the initial data

Finally (25) is derived formally with respect to t and thesame procedure is repeated as in the first estimate but for thenonlinear convective terms Now in order to bound theinitial data for vt initial data in W22 is chosen

ΔΠt minus βΠtz minuseβz

Prnabla middot vt middot nablav + ϕtv + v middot nablavt + ϕvt( 1113857

+Raeβzτtz

1Pr

zvt

zt+ vt middot nablav + ϕtv + v middot nablavt + ϕvt1113888 1113889 minus Δvt

minus nabla eminus βzΠt( 1113857 minus βeminus βzΠtk + Raτtk

zτt

zt+ vt middot nablaτ + v middot nablaτt minus Δτt v

zt

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(72)

Again the terms depending on ϕ or ϕt will be boundedexactly as the corresponding convective ones

Lemma 5 Let us define the energy function as

E(t) ≔12

vt

22

Pr+ Ra τt

22

⎛⎝ ⎞⎠ (73)

en for sufficiently small εgt 0 the following inequalityholds truedEdt

(t) + c nablavt

22 + Ra nablaτt

221113874 1113875leC(β PrRa)Δv

22E(t)

+ C(βPr)Raε

vt

22

(74)

where c isin (0 1)

Proof Let us start with the equation

ddt

E(t) + nablavt

22 + Ra nablaτt

22 minus e

minus βznablaΠt vt1113872 1113873 + 2Ra τt vzt( 1113857 +

minus1Pr

vt middot nablav + v middot nablavt vt( 1113857 minus1Pr

ϕtv + ϕvt vt( 1113857 minus Ra vt middot nablaτ + v middot nablaτt τt( 1113857

(75)

Technically the estimates of the nonlinear convectiveterms are the only different ones from the previous onessince the terms are not homogeneous

eminus βznablaΠt vt1113872 1113873

11138681113868111386811138681113868τt0le c vt

2 vt middot nablav

2 + v middot nablavt

21113872 1113873

le c vt

2 vt

3nablav6 + vt

2vinfin nablavt

21113872 1113873

le c vt

322 nablavt

122 Δv2 + vt

2Δv2 nablavt

21113874 1113875

le c Δv22 vt

22 + ε nablavt

221113874 1113875

(76)

e same estimates can be repeated for the other non-linear terms such as

ϕvt vt( 1113857le cnablav6 vt

2 vt

3

ϕtv vt( 1113857le c vt

2vinfin nablavt

2

vt middot nablaτ τt( 1113857le cnablaτ6 vt

2 τt

3

v middot nablaτt τt( 1113857le c τt

2vinfin nablaτt

2

(77)

e gradient terms can be absorbed at the left-handside

Starting from (74) where the positive term at the left-hand side can be neglected and the term vt

22 at the right-

International Journal of Differential Equations 9

hand side can be bounded by E(t) by a generalizedGronwall argument one shows that

E(t)le ec 1113938

T

0Δv(t)22+1( )dt

E(0) (78)

As a matter of fact (71) implies v isin L2(0 T W22(Ω0))so thatE(t) is bounded in (0 T) Further by coming back to(74) and integrating again it follows also

1113946T

0nablavt

22 + nablaτt

221113874 1113875dtleO v0

22 τ0

221113872 1113873 (79)

and this is independent of the size of Ra

Theorem 1 Let βlt 2π and Pr and Ra be arbitrary thensystem (25) with initial data (v0 τ0) isinW22(Ω0) admits asolution a la Ladyzhenskaya namely

(v τ) isin Linfin 0 T W

22 Ω0( 11138571113872 1113873capL2 0 T W

32 Ω0( 11138571113872 1113873

vt τt( 1113857 isin Linfin 0 T L

2 Ω0( 11138571113872 1113873cap L2 0 T W

12 Ω0( 11138571113872 1113873(80)

where

T ≔ logE2(0) + 1

E2(0)1113888 1113889

14C

(81)

Moreover if nabla middot v0 0 then nabla middot v(t) 0 for all t isin (0 T]Finally if Ra v022 and τ022 are sufficiently smalldepending on Pr and β then T infin is taken in (80) and (v τ)

decays exponentially fast to zero

Proof Employing the estimates proved in Lemma 4 it isroutine to show that the approximate solutions constructedwith the Galerkin method converge to a solution

(v τ) isin Linfin 0 T W

12 Ω0( 11138571113872 1113873capL2 0 T W

22 Ω0( 11138571113872 1113873

Π isin L2 0 T W

22 Ω0( 11138571113872 1113873

vt τt( 1113857 isin Linfin 0 T L

2 Ω0( 11138571113872 1113873cap L2 0 T W

12 Ω0( 11138571113872 1113873

(82)

where T is given in (81)e proof of the further regularity properties implying

that the solution fulfils system (17) too is based on (78) and(79) and in particular on the uniform boundedness of vt2as well as the integrability of nablavt(t)22 To this end viewingthe momentum balance as a Poisson problem and increasingthe pressure in the usual way the following is obtained withthe help of Lemma 1

v(t)22 leC v middot nablav2 +nablaΠ2 + vt

2 +τ21113872 1113873

leC nablav42 + εΔv

22 + vt

2 +τ21113872 1113873

(83)

e ε term at the right-hand side can be absorbed by theleft-hand side so that finally the following bound is obtained

ess sup(0T)

v(t)22 leCess sup(0T)

nablav(t)42 +τ2 + vt

21113872 1113873

leO v0

22 τ0

221113872 1113873

(84)

Moreover again from Lemma 1 it is also obtained that

1113946T

0v(t)

232dtleC 1113946

T

0v middot nablav

212 +Π

222 + vt

212 +τ

2121113874 1113875dt

(85)

It is remarked that as noticed in Section 2 the com-patibility condition related to the Neumann problem isverified Here the nonlinear terms at the right-hand side canbe increased by using (14) and interpolation inequality

v middot D2v

2 le cvinfin D

2v

2 le c D2v

22

nablav(nablav)T

le cnablav

24 le cnablav6nablav3 le c D

2v

22

(86)

e proof that τ is bounded in Linfin(0 T W12(Ω0))capL2(0 T W22(Ω0)) is similar and simpler

e proof of global existence follows from Lemmas 3 and5 by setting

E0(t) ≔ E(t) + E(t)

P0(t) ≔ nablav(t)22 +Δv(t)

22 + nablavt(t)

22

+ Ra nablaτ(t)22 +Δτ(t)

22 + nablaτt(t)

221113874 1113875

(87)

In fact if inequalities (49) (51) and (74) are addedbringing to the left-hand side the three terms with factorRaε and using the Poincare inequality

Ele kP (88)

to absorb them in P0 for sufficiently small Ra (dependingon ε) one gets

ddt

E(t) + c1P(t)le c2 E32

(t) + E3(t)1113872 1113873 (89)

e constant c1 is now fixed positive by suitablyrestricting Ra Again using the Poincare inequality on theleft-hand side of (89) one infers

ddt

E(t) + c1 minus c2 E12

(t) + E2(t)1113872 11138731113960 1113961

1k

E(t)le 0 (90)

us the exponential decay follows by choosing at theinitial time

E12

(0) + E2(0)le

12

c1

c2 (91)

In fact by continuity for some time E(t)leE(0) Butthis reinforces the previous condition so that the positivefunction in square brackets cannot decrease and is boundedfrom below by a positive constant for all times

Data Availability

e data that support the findings of this study are availablewithin the article

Conflicts of Interest

e author declares that there are no conflicts of interest

10 International Journal of Differential Equations

References

[1] A Giacobbe and G Mulone ldquoStability in the rotating Benardproblem and its optimal Lyapunov functionsrdquo Acta Appli-candae Mathematicae vol 132 no 1 2014

[2] A Passerini C Ferrario M Ruzicka and G ater ldquoe-oretical results on steady convective flows between horizontalcoaxial cylindersrdquo SIAM Journal on Applied Mathematicsvol 71 no 2 pp 465ndash486 2011

[3] A Passerini and C Ferrario ldquoA theoretical study of the firsttransition for the non-linear Stokes problem in a horizontalannulusrdquo International Journal of Non-linear Mechanicsvol 78 no 1 pp 1ndash8 2015

[4] A De Martino and A Passerini ldquoExistence and nonlinearstability of convective solutions for almost compressible fluidsin Benard problemrdquo Journal of Mathematical Physics vol 60no 11 Article ID 113101 2019

[5] F Crispo and P Maremonti ldquoA high regularity result ofsolutions to modified p-Stokes equationsrdquo Nonlinear Anal-ysis vol 118 pp 97ndash129 2015

[6] H Gouin and T Ruggeri ldquoA consistent thermodynamicalmodel of incompressible media as limit case of quasi-thermal-incompressible materialsrdquo International Journal of Non-Linear Mechanics vol 47 no 6 pp 688ndash693 2012

[7] A Passerini and T Ruggeri ldquoe Benard problem for quasi-thermal-incompressible materials a linear analysisrdquo Inter-national Journal of Non-Linear Mechanics vol 67 no 1pp 178ndash185 2014

[8] S Gatti and E Vuk ldquoSingular limit of equations for linearviscoelastic fluids with periodic boundary conditionsrdquo In-ternational Journal of Non-Linear Mechanics vol 41 no 4pp 518ndash526 2006

[9] A Corli and A Passerini ldquoe Benard problem for slightlycompressible materials existence and linear instabilityrdquoMediterranean Journal of Mathematics vol 16 no 1 2019

[10] G P Galdi ldquoNon-linear stability of the magnetic Benardproblem via a generalized energy methodrdquo Archive for Ra-tional Mechanics and Analysis vol 87 no 2 pp 167ndash1861985

[11] G P Galdi and M Padula ldquoA new approach to energy theoryin the stability of fluid motionrdquo Archive for Rational Me-chanics and Analysis vol 110 no 3 pp 187ndash286 1990

[12] A De Martino and A Passerini ldquoA Lorenz model for almostcompressible fluidsrdquo Mediterranean Journal of Mathematicsvol 17 no 1 p 12 2020

[13] G P Galdi An Introduction to the Mathematical eory of theNavier-Stokes Equation Springer New York NY USA 1998

[14] O Ladyzhenskaya e Mathematical eory of Viscous In-compressible Fluids Gordon amp Breach London UK 1963

[15] G Iooss ldquoeorie non lineaire de la stabilite des ecoulementslaminaires dans le cas de ldquolrsquoechange des stabilitesrdquo Archive forRational Mechanics and Analysis vol 40 no 3 pp 166ndash2081971

[16] M Schechter ldquoCoerciveness in Lprdquo Transactions of theAmerican Mathematical Society vol 107 no 1 pp 10ndash291963

International Journal of Differential Equations 11

the energy is boundedNext coming back to (65) and integrating in [0 T) one

deduces a bound for the norm in Linfin(0 T W12(Ω0))capL2(0 T W22(Ω0)) as can be seen hereafter the inequality

1113946T

0P(t)dtleC log

E2(0) + 1E2(0)

1113888 1113889

14

E(0) + E12

(0)1113872 1113873

(71)

implies in particular the integrability of nablav(t)22 and ofΔv(t)22

Although the Galerkin solutions exist for all tgt 0 theiruniform estimates exist only in the finite interval defined in(69) Global-in-time solution will be found only by askingsuitable smallness of Ra and of the initial data

Finally (25) is derived formally with respect to t and thesame procedure is repeated as in the first estimate but for thenonlinear convective terms Now in order to bound theinitial data for vt initial data in W22 is chosen

ΔΠt minus βΠtz minuseβz

Prnabla middot vt middot nablav + ϕtv + v middot nablavt + ϕvt( 1113857

+Raeβzτtz

1Pr

zvt

zt+ vt middot nablav + ϕtv + v middot nablavt + ϕvt1113888 1113889 minus Δvt

minus nabla eminus βzΠt( 1113857 minus βeminus βzΠtk + Raτtk

zτt

zt+ vt middot nablaτ + v middot nablaτt minus Δτt v

zt

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(72)

Again the terms depending on ϕ or ϕt will be boundedexactly as the corresponding convective ones

Lemma 5 Let us define the energy function as

E(t) ≔12

vt

22

Pr+ Ra τt

22

⎛⎝ ⎞⎠ (73)

en for sufficiently small εgt 0 the following inequalityholds truedEdt

(t) + c nablavt

22 + Ra nablaτt

221113874 1113875leC(β PrRa)Δv

22E(t)

+ C(βPr)Raε

vt

22

(74)

where c isin (0 1)

Proof Let us start with the equation

ddt

E(t) + nablavt

22 + Ra nablaτt

22 minus e

minus βznablaΠt vt1113872 1113873 + 2Ra τt vzt( 1113857 +

minus1Pr

vt middot nablav + v middot nablavt vt( 1113857 minus1Pr

ϕtv + ϕvt vt( 1113857 minus Ra vt middot nablaτ + v middot nablaτt τt( 1113857

(75)

Technically the estimates of the nonlinear convectiveterms are the only different ones from the previous onessince the terms are not homogeneous

eminus βznablaΠt vt1113872 1113873

11138681113868111386811138681113868τt0le c vt

2 vt middot nablav

2 + v middot nablavt

21113872 1113873

le c vt

2 vt

3nablav6 + vt

2vinfin nablavt

21113872 1113873

le c vt

322 nablavt

122 Δv2 + vt

2Δv2 nablavt

21113874 1113875

le c Δv22 vt

22 + ε nablavt

221113874 1113875

(76)

e same estimates can be repeated for the other non-linear terms such as

ϕvt vt( 1113857le cnablav6 vt

2 vt

3

ϕtv vt( 1113857le c vt

2vinfin nablavt

2

vt middot nablaτ τt( 1113857le cnablaτ6 vt

2 τt

3

v middot nablaτt τt( 1113857le c τt

2vinfin nablaτt

2

(77)

e gradient terms can be absorbed at the left-handside

Starting from (74) where the positive term at the left-hand side can be neglected and the term vt

22 at the right-

International Journal of Differential Equations 9

hand side can be bounded by E(t) by a generalizedGronwall argument one shows that

E(t)le ec 1113938

T

0Δv(t)22+1( )dt

E(0) (78)

As a matter of fact (71) implies v isin L2(0 T W22(Ω0))so thatE(t) is bounded in (0 T) Further by coming back to(74) and integrating again it follows also

1113946T

0nablavt

22 + nablaτt

221113874 1113875dtleO v0

22 τ0

221113872 1113873 (79)

and this is independent of the size of Ra

Theorem 1 Let βlt 2π and Pr and Ra be arbitrary thensystem (25) with initial data (v0 τ0) isinW22(Ω0) admits asolution a la Ladyzhenskaya namely

(v τ) isin Linfin 0 T W

22 Ω0( 11138571113872 1113873capL2 0 T W

32 Ω0( 11138571113872 1113873

vt τt( 1113857 isin Linfin 0 T L

2 Ω0( 11138571113872 1113873cap L2 0 T W

12 Ω0( 11138571113872 1113873(80)

where

T ≔ logE2(0) + 1

E2(0)1113888 1113889

14C

(81)

Moreover if nabla middot v0 0 then nabla middot v(t) 0 for all t isin (0 T]Finally if Ra v022 and τ022 are sufficiently smalldepending on Pr and β then T infin is taken in (80) and (v τ)

decays exponentially fast to zero

Proof Employing the estimates proved in Lemma 4 it isroutine to show that the approximate solutions constructedwith the Galerkin method converge to a solution

(v τ) isin Linfin 0 T W

12 Ω0( 11138571113872 1113873capL2 0 T W

22 Ω0( 11138571113872 1113873

Π isin L2 0 T W

22 Ω0( 11138571113872 1113873

vt τt( 1113857 isin Linfin 0 T L

2 Ω0( 11138571113872 1113873cap L2 0 T W

12 Ω0( 11138571113872 1113873

(82)

where T is given in (81)e proof of the further regularity properties implying

that the solution fulfils system (17) too is based on (78) and(79) and in particular on the uniform boundedness of vt2as well as the integrability of nablavt(t)22 To this end viewingthe momentum balance as a Poisson problem and increasingthe pressure in the usual way the following is obtained withthe help of Lemma 1

v(t)22 leC v middot nablav2 +nablaΠ2 + vt

2 +τ21113872 1113873

leC nablav42 + εΔv

22 + vt

2 +τ21113872 1113873

(83)

e ε term at the right-hand side can be absorbed by theleft-hand side so that finally the following bound is obtained

ess sup(0T)

v(t)22 leCess sup(0T)

nablav(t)42 +τ2 + vt

21113872 1113873

leO v0

22 τ0

221113872 1113873

(84)

Moreover again from Lemma 1 it is also obtained that

1113946T

0v(t)

232dtleC 1113946

T

0v middot nablav

212 +Π

222 + vt

212 +τ

2121113874 1113875dt

(85)

It is remarked that as noticed in Section 2 the com-patibility condition related to the Neumann problem isverified Here the nonlinear terms at the right-hand side canbe increased by using (14) and interpolation inequality

v middot D2v

2 le cvinfin D

2v

2 le c D2v

22

nablav(nablav)T

le cnablav

24 le cnablav6nablav3 le c D

2v

22

(86)

e proof that τ is bounded in Linfin(0 T W12(Ω0))capL2(0 T W22(Ω0)) is similar and simpler

e proof of global existence follows from Lemmas 3 and5 by setting

E0(t) ≔ E(t) + E(t)

P0(t) ≔ nablav(t)22 +Δv(t)

22 + nablavt(t)

22

+ Ra nablaτ(t)22 +Δτ(t)

22 + nablaτt(t)

221113874 1113875

(87)

In fact if inequalities (49) (51) and (74) are addedbringing to the left-hand side the three terms with factorRaε and using the Poincare inequality

Ele kP (88)

to absorb them in P0 for sufficiently small Ra (dependingon ε) one gets

ddt

E(t) + c1P(t)le c2 E32

(t) + E3(t)1113872 1113873 (89)

e constant c1 is now fixed positive by suitablyrestricting Ra Again using the Poincare inequality on theleft-hand side of (89) one infers

ddt

E(t) + c1 minus c2 E12

(t) + E2(t)1113872 11138731113960 1113961

1k

E(t)le 0 (90)

us the exponential decay follows by choosing at theinitial time

E12

(0) + E2(0)le

12

c1

c2 (91)

In fact by continuity for some time E(t)leE(0) Butthis reinforces the previous condition so that the positivefunction in square brackets cannot decrease and is boundedfrom below by a positive constant for all times

Data Availability

e data that support the findings of this study are availablewithin the article

Conflicts of Interest

e author declares that there are no conflicts of interest

10 International Journal of Differential Equations

References

[1] A Giacobbe and G Mulone ldquoStability in the rotating Benardproblem and its optimal Lyapunov functionsrdquo Acta Appli-candae Mathematicae vol 132 no 1 2014

[2] A Passerini C Ferrario M Ruzicka and G ater ldquoe-oretical results on steady convective flows between horizontalcoaxial cylindersrdquo SIAM Journal on Applied Mathematicsvol 71 no 2 pp 465ndash486 2011

[3] A Passerini and C Ferrario ldquoA theoretical study of the firsttransition for the non-linear Stokes problem in a horizontalannulusrdquo International Journal of Non-linear Mechanicsvol 78 no 1 pp 1ndash8 2015

[4] A De Martino and A Passerini ldquoExistence and nonlinearstability of convective solutions for almost compressible fluidsin Benard problemrdquo Journal of Mathematical Physics vol 60no 11 Article ID 113101 2019

[5] F Crispo and P Maremonti ldquoA high regularity result ofsolutions to modified p-Stokes equationsrdquo Nonlinear Anal-ysis vol 118 pp 97ndash129 2015

[6] H Gouin and T Ruggeri ldquoA consistent thermodynamicalmodel of incompressible media as limit case of quasi-thermal-incompressible materialsrdquo International Journal of Non-Linear Mechanics vol 47 no 6 pp 688ndash693 2012

[7] A Passerini and T Ruggeri ldquoe Benard problem for quasi-thermal-incompressible materials a linear analysisrdquo Inter-national Journal of Non-Linear Mechanics vol 67 no 1pp 178ndash185 2014

[8] S Gatti and E Vuk ldquoSingular limit of equations for linearviscoelastic fluids with periodic boundary conditionsrdquo In-ternational Journal of Non-Linear Mechanics vol 41 no 4pp 518ndash526 2006

[9] A Corli and A Passerini ldquoe Benard problem for slightlycompressible materials existence and linear instabilityrdquoMediterranean Journal of Mathematics vol 16 no 1 2019

[10] G P Galdi ldquoNon-linear stability of the magnetic Benardproblem via a generalized energy methodrdquo Archive for Ra-tional Mechanics and Analysis vol 87 no 2 pp 167ndash1861985

[11] G P Galdi and M Padula ldquoA new approach to energy theoryin the stability of fluid motionrdquo Archive for Rational Me-chanics and Analysis vol 110 no 3 pp 187ndash286 1990

[12] A De Martino and A Passerini ldquoA Lorenz model for almostcompressible fluidsrdquo Mediterranean Journal of Mathematicsvol 17 no 1 p 12 2020

[13] G P Galdi An Introduction to the Mathematical eory of theNavier-Stokes Equation Springer New York NY USA 1998

[14] O Ladyzhenskaya e Mathematical eory of Viscous In-compressible Fluids Gordon amp Breach London UK 1963

[15] G Iooss ldquoeorie non lineaire de la stabilite des ecoulementslaminaires dans le cas de ldquolrsquoechange des stabilitesrdquo Archive forRational Mechanics and Analysis vol 40 no 3 pp 166ndash2081971

[16] M Schechter ldquoCoerciveness in Lprdquo Transactions of theAmerican Mathematical Society vol 107 no 1 pp 10ndash291963

International Journal of Differential Equations 11

hand side can be bounded by E(t) by a generalizedGronwall argument one shows that

E(t)le ec 1113938

T

0Δv(t)22+1( )dt

E(0) (78)

As a matter of fact (71) implies v isin L2(0 T W22(Ω0))so thatE(t) is bounded in (0 T) Further by coming back to(74) and integrating again it follows also

1113946T

0nablavt

22 + nablaτt

221113874 1113875dtleO v0

22 τ0

221113872 1113873 (79)

and this is independent of the size of Ra

Theorem 1 Let βlt 2π and Pr and Ra be arbitrary thensystem (25) with initial data (v0 τ0) isinW22(Ω0) admits asolution a la Ladyzhenskaya namely

(v τ) isin Linfin 0 T W

22 Ω0( 11138571113872 1113873capL2 0 T W

32 Ω0( 11138571113872 1113873

vt τt( 1113857 isin Linfin 0 T L

2 Ω0( 11138571113872 1113873cap L2 0 T W

12 Ω0( 11138571113872 1113873(80)

where

T ≔ logE2(0) + 1

E2(0)1113888 1113889

14C

(81)

Moreover if nabla middot v0 0 then nabla middot v(t) 0 for all t isin (0 T]Finally if Ra v022 and τ022 are sufficiently smalldepending on Pr and β then T infin is taken in (80) and (v τ)

decays exponentially fast to zero

Proof Employing the estimates proved in Lemma 4 it isroutine to show that the approximate solutions constructedwith the Galerkin method converge to a solution

(v τ) isin Linfin 0 T W

12 Ω0( 11138571113872 1113873capL2 0 T W

22 Ω0( 11138571113872 1113873

Π isin L2 0 T W

22 Ω0( 11138571113872 1113873

vt τt( 1113857 isin Linfin 0 T L

2 Ω0( 11138571113872 1113873cap L2 0 T W

12 Ω0( 11138571113872 1113873

(82)

where T is given in (81)e proof of the further regularity properties implying

that the solution fulfils system (17) too is based on (78) and(79) and in particular on the uniform boundedness of vt2as well as the integrability of nablavt(t)22 To this end viewingthe momentum balance as a Poisson problem and increasingthe pressure in the usual way the following is obtained withthe help of Lemma 1

v(t)22 leC v middot nablav2 +nablaΠ2 + vt

2 +τ21113872 1113873

leC nablav42 + εΔv

22 + vt

2 +τ21113872 1113873

(83)

e ε term at the right-hand side can be absorbed by theleft-hand side so that finally the following bound is obtained

ess sup(0T)

v(t)22 leCess sup(0T)

nablav(t)42 +τ2 + vt

21113872 1113873

leO v0

22 τ0

221113872 1113873

(84)

Moreover again from Lemma 1 it is also obtained that

1113946T

0v(t)

232dtleC 1113946

T

0v middot nablav

212 +Π

222 + vt

212 +τ

2121113874 1113875dt

(85)

It is remarked that as noticed in Section 2 the com-patibility condition related to the Neumann problem isverified Here the nonlinear terms at the right-hand side canbe increased by using (14) and interpolation inequality

v middot D2v

2 le cvinfin D

2v

2 le c D2v

22

nablav(nablav)T

le cnablav

24 le cnablav6nablav3 le c D

2v

22

(86)

e proof that τ is bounded in Linfin(0 T W12(Ω0))capL2(0 T W22(Ω0)) is similar and simpler

e proof of global existence follows from Lemmas 3 and5 by setting

E0(t) ≔ E(t) + E(t)

P0(t) ≔ nablav(t)22 +Δv(t)

22 + nablavt(t)

22

+ Ra nablaτ(t)22 +Δτ(t)

22 + nablaτt(t)

221113874 1113875

(87)

In fact if inequalities (49) (51) and (74) are addedbringing to the left-hand side the three terms with factorRaε and using the Poincare inequality

Ele kP (88)

to absorb them in P0 for sufficiently small Ra (dependingon ε) one gets

ddt

E(t) + c1P(t)le c2 E32

(t) + E3(t)1113872 1113873 (89)

e constant c1 is now fixed positive by suitablyrestricting Ra Again using the Poincare inequality on theleft-hand side of (89) one infers

ddt

E(t) + c1 minus c2 E12

(t) + E2(t)1113872 11138731113960 1113961

1k

E(t)le 0 (90)

us the exponential decay follows by choosing at theinitial time

E12

(0) + E2(0)le

12

c1

c2 (91)

In fact by continuity for some time E(t)leE(0) Butthis reinforces the previous condition so that the positivefunction in square brackets cannot decrease and is boundedfrom below by a positive constant for all times

Data Availability

e data that support the findings of this study are availablewithin the article

Conflicts of Interest

e author declares that there are no conflicts of interest

10 International Journal of Differential Equations

References

[1] A Giacobbe and G Mulone ldquoStability in the rotating Benardproblem and its optimal Lyapunov functionsrdquo Acta Appli-candae Mathematicae vol 132 no 1 2014

[2] A Passerini C Ferrario M Ruzicka and G ater ldquoe-oretical results on steady convective flows between horizontalcoaxial cylindersrdquo SIAM Journal on Applied Mathematicsvol 71 no 2 pp 465ndash486 2011

[3] A Passerini and C Ferrario ldquoA theoretical study of the firsttransition for the non-linear Stokes problem in a horizontalannulusrdquo International Journal of Non-linear Mechanicsvol 78 no 1 pp 1ndash8 2015

[4] A De Martino and A Passerini ldquoExistence and nonlinearstability of convective solutions for almost compressible fluidsin Benard problemrdquo Journal of Mathematical Physics vol 60no 11 Article ID 113101 2019

[5] F Crispo and P Maremonti ldquoA high regularity result ofsolutions to modified p-Stokes equationsrdquo Nonlinear Anal-ysis vol 118 pp 97ndash129 2015

[6] H Gouin and T Ruggeri ldquoA consistent thermodynamicalmodel of incompressible media as limit case of quasi-thermal-incompressible materialsrdquo International Journal of Non-Linear Mechanics vol 47 no 6 pp 688ndash693 2012

[7] A Passerini and T Ruggeri ldquoe Benard problem for quasi-thermal-incompressible materials a linear analysisrdquo Inter-national Journal of Non-Linear Mechanics vol 67 no 1pp 178ndash185 2014

[8] S Gatti and E Vuk ldquoSingular limit of equations for linearviscoelastic fluids with periodic boundary conditionsrdquo In-ternational Journal of Non-Linear Mechanics vol 41 no 4pp 518ndash526 2006

[9] A Corli and A Passerini ldquoe Benard problem for slightlycompressible materials existence and linear instabilityrdquoMediterranean Journal of Mathematics vol 16 no 1 2019

[10] G P Galdi ldquoNon-linear stability of the magnetic Benardproblem via a generalized energy methodrdquo Archive for Ra-tional Mechanics and Analysis vol 87 no 2 pp 167ndash1861985

[11] G P Galdi and M Padula ldquoA new approach to energy theoryin the stability of fluid motionrdquo Archive for Rational Me-chanics and Analysis vol 110 no 3 pp 187ndash286 1990

[12] A De Martino and A Passerini ldquoA Lorenz model for almostcompressible fluidsrdquo Mediterranean Journal of Mathematicsvol 17 no 1 p 12 2020

[13] G P Galdi An Introduction to the Mathematical eory of theNavier-Stokes Equation Springer New York NY USA 1998

[14] O Ladyzhenskaya e Mathematical eory of Viscous In-compressible Fluids Gordon amp Breach London UK 1963

[15] G Iooss ldquoeorie non lineaire de la stabilite des ecoulementslaminaires dans le cas de ldquolrsquoechange des stabilitesrdquo Archive forRational Mechanics and Analysis vol 40 no 3 pp 166ndash2081971

[16] M Schechter ldquoCoerciveness in Lprdquo Transactions of theAmerican Mathematical Society vol 107 no 1 pp 10ndash291963

International Journal of Differential Equations 11

References

[1] A Giacobbe and G Mulone ldquoStability in the rotating Benardproblem and its optimal Lyapunov functionsrdquo Acta Appli-candae Mathematicae vol 132 no 1 2014

[2] A Passerini C Ferrario M Ruzicka and G ater ldquoe-oretical results on steady convective flows between horizontalcoaxial cylindersrdquo SIAM Journal on Applied Mathematicsvol 71 no 2 pp 465ndash486 2011

[3] A Passerini and C Ferrario ldquoA theoretical study of the firsttransition for the non-linear Stokes problem in a horizontalannulusrdquo International Journal of Non-linear Mechanicsvol 78 no 1 pp 1ndash8 2015

[4] A De Martino and A Passerini ldquoExistence and nonlinearstability of convective solutions for almost compressible fluidsin Benard problemrdquo Journal of Mathematical Physics vol 60no 11 Article ID 113101 2019

[5] F Crispo and P Maremonti ldquoA high regularity result ofsolutions to modified p-Stokes equationsrdquo Nonlinear Anal-ysis vol 118 pp 97ndash129 2015

[6] H Gouin and T Ruggeri ldquoA consistent thermodynamicalmodel of incompressible media as limit case of quasi-thermal-incompressible materialsrdquo International Journal of Non-Linear Mechanics vol 47 no 6 pp 688ndash693 2012

[7] A Passerini and T Ruggeri ldquoe Benard problem for quasi-thermal-incompressible materials a linear analysisrdquo Inter-national Journal of Non-Linear Mechanics vol 67 no 1pp 178ndash185 2014

[8] S Gatti and E Vuk ldquoSingular limit of equations for linearviscoelastic fluids with periodic boundary conditionsrdquo In-ternational Journal of Non-Linear Mechanics vol 41 no 4pp 518ndash526 2006

[9] A Corli and A Passerini ldquoe Benard problem for slightlycompressible materials existence and linear instabilityrdquoMediterranean Journal of Mathematics vol 16 no 1 2019

[10] G P Galdi ldquoNon-linear stability of the magnetic Benardproblem via a generalized energy methodrdquo Archive for Ra-tional Mechanics and Analysis vol 87 no 2 pp 167ndash1861985

[11] G P Galdi and M Padula ldquoA new approach to energy theoryin the stability of fluid motionrdquo Archive for Rational Me-chanics and Analysis vol 110 no 3 pp 187ndash286 1990

[12] A De Martino and A Passerini ldquoA Lorenz model for almostcompressible fluidsrdquo Mediterranean Journal of Mathematicsvol 17 no 1 p 12 2020

[13] G P Galdi An Introduction to the Mathematical eory of theNavier-Stokes Equation Springer New York NY USA 1998

[14] O Ladyzhenskaya e Mathematical eory of Viscous In-compressible Fluids Gordon amp Breach London UK 1963

[15] G Iooss ldquoeorie non lineaire de la stabilite des ecoulementslaminaires dans le cas de ldquolrsquoechange des stabilitesrdquo Archive forRational Mechanics and Analysis vol 40 no 3 pp 166ndash2081971

[16] M Schechter ldquoCoerciveness in Lprdquo Transactions of theAmerican Mathematical Society vol 107 no 1 pp 10ndash291963

International Journal of Differential Equations 11