# Bénard Problem for Slightly Compressible Fluids: Existence and Nonlinear Stability in 3D

Authors: Arianna Passerini
Journal: International Journal of Differential Equations
 (2020)

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## Abstract

This paper shows the existence, uniqueness, and asymptotic behavior in time of regular solutions (a la Ladyzhenskaya) to the Bénard problem for a heat-conducting fluid model generalizing the classical Oberbeck–Boussinesq one. The novelty of this model, introduced by Corli and Passerini, 2019, and Passerini and Ruggeri, 2014, consists in allowing the density of the fluid to also depend on the pressure field, which, as shown by Passerini and Ruggeri, 2014, is a necessary request from a thermodynamic viewpoint when dealing with convective problems. This property adds to the problem a rather interesting mathematical challenge that is not encountered in the classical model, thus requiring a new approach for its resolution.


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## Body

## 1. Introduction

As is well known, one of the most accepted models in the study of convection problems in fluid mechanics is the so-called Oberbeck–Boussinesq approximation (hereafter denoted by O–B) [1–5]. The latter is characterized by the fact that one keeps the fluid incompressible and allows for density changes only in the buoyancy term of the linear momentum equation, by assuming a linear dependence on temperature.However, as observed in [6] and further elaborated in [7], O–B may not be entirely satisfactory or even consistent from the physical viewpoint [8] if the density is not allowed to have a dependence also on the pressure field p. As a result, in [9], a more general theory aimed at removing this incongruence was proposed. More specifically, while still keeping the incompressibility constraint, the density in the buoyancy term was allowed to depend, linearly, also on p. The remarkable feature of this new model is that now, unlike the classical O–B, the linear momentum equation contains a pressure term that is not in the classical gradient form. From a mathematical perspective, this feature is very challenging in that some fundamental estimates necessary to ensure stability properties and well-posedness of the relevant problems [10, 11] no longer hold and, consequently, the study of such problems requires fresh ideas and different approach.With this in mind, a systematic study of the mathematical properties of the new model was started, focusing on the classical Bénard problem. As is well known, the latter regards convective motions stemming from a motionless basic flow,r0, of a horizontal layer of fluid heated from below. For the new model, the state r0 is given by [7](1)v≡0,Tz≡Trz=T0−δThz,pz≡prz=p0+pbe−ρ0gβz+1β2αδTρ0gh1−e−ρ0gβz−1β1−e−ρ0gβz+αδThz,with the z coordinate in the vertical direction. Here, v, T, and p are the velocity, temperature, and pressure fields, respectively, ρ0 and p0 are the (constant) gauge density and pressure, T0 and T0+δT are the temperature at bottom and top of the layer, and h is the thickness of the layer. Finally, α and β are the positive material constants. Notice that (1) tends to the motionless solution for the same problem in the O–B approximation by letting β⟶0. In [4, 9, 12], the linear and nonlinear stability of r0, as well as the existence and uniqueness for the associated “perturbed problem,” was investigated. All this work was done in the case of the plane flow.The objective of the current paper is to investigate the same problem in the more general and appropriate three-dimensional setting. The approach followed here to overcome the problem of the pressure mentioned earlier on is to introduce a new equation of the elliptic type for the pressure field, disregarding the incompressibility constraint (for steady and for nonlinear equations close to the one of non-Newtonian fluids, this approach that allows to achieve a posteriori the divergence-free equation was considered in [5]. Actually, although this work is done in a “parabolic context” (for the kinetic field), the technique in [5] in some sense has inspired the approach in the present paper). In such a way, the original system of equations is converted to three highly nonlinearly coupled equations for velocity, temperature, and pressure fields. However, for this new problem, it is shown that if the velocity field, v, is sufficiently smooth and also solenoidal at time t=0, then v must be solenoidal in the entire interval of time where the solution exists, thus providing a solution to the original problem. In this way, it is shown that, if β<2π, the relevant perturbed problem (to the state r0) has a local, strong solution (a la Ladyzhenskaya) corresponding to initial data of arbitrary size (in a suitable Sobolev class). Such a solution becomes, in fact, global if this size is appropriately restricted and the Rayleigh number is below a certain value, see Theorem 1.The plan of the paper is given in the following. After formulating the problem and collecting some preliminary results in Section1, in Section 2, the classical Galerkin method is used (see, for instance, [4]) to construct an “approximate” solution. In Section 3, which constitutes the main core of this paper, a number of fundamental estimates on the approximate solution is proven that, again by classical techniques, allows us to conclude that the latter converges to a solution of the original problem having the properties described above.


## 2. Notations and Preliminary Results

The flow domain is an infinite, horizontal layer bounded in the verticalz-direction with thickness h and unbounded and invariant in the x and y directions. It is assumed that the generic perturbed field is periodic in x and y of a given period Lx and Ly, respectively, an assumption that is justified by the fact that the convective roll pattern has a periodic structure. The relevant spatial region thus becomes a “periodicity cell” defined (in nondimensional form) as(2)Ω0=0,a×0,b×0,1,with a=Lx/h and b=Ly/h. Just for simplicity, set, herein, a=b=1. The mean value of a function f is indicated as(3)f=1Ω0∫Ω0fx,y,z,tdxdydz,and a partial derivative with a subscript ∂ξϕ=∂ϕ/∂ξ=ϕξ with ξ∈x,y,z,t, and similarly for higher-order derivatives. As customary, the inner product in L2Ω0 is denoted by(4)u,v≔∫Ω0u⋅vdxdydz,where u and v denote the scalar vector or tensor fields. The components of a vector u are denoted by u≡ux,uy,uz. Let Wm,pΩ0 be the usual Sobolev space and utm,p be the associated norm. The subspace of Wm,pΩ0 constituted by those u with u=0 is denoted by Wm,pΩ0. By L∞0,T;Wm,pΩ0, we denote the space of functions u such that(5)u∞,m,p≔esssupt∈0,Tutm,p<∞.Likewise, forq∈1,∞, the Bochner space Lq0,T;Wm,pΩ0 is the space defined by the norm(6)uq,m,p≔∫0Tutm,pqdx1/q<∞.A decisive role in our stability analysis is played by the Poincaré inequality inΩ0:(7)u2≤C∇u2.As is known (see [13]), set Γ=Ω¯0∩z=0,1; this inequality is true if at least one of the following cases is satisfied: (a) functions with a zero mean value, (b) vector functions with a vanishing normal component on Γ, and (c) scalar functions vanishing on Γ. Throughout, the same symbol C is used to denote the involved constant, which depends only on the domain. Since boundary conditions are used for which the following integration by parts is valid:(8)∇u2=−Δu,u,if the Poincaré inequality holds true for u, then by the Schwarz inequality, it also follows(9)∇u2≤CΔu2.As a matter of fact, inΩ0, there is an equivalence between the norm of the Laplacian and that of the full set of the second derivatives, as shown by Ladyzhenskaya (see [14], equations (17) and (18) at page 18]):(10)Δu2=D2u2.Further tools, herein suitable, are some typical 3D immersion inequalities: the Sobolev–Gagliardo–Nirenberg inequality:(11)u6≤Cu1,2,which in case the Poincaré inequality also holds true becomes(12)u6≤C∇u2,(13)∇u6≤CΔu2.In what follows, Morrey’s inequality is also used (still simplified by the Poincaré inequality):(14)u∞≤C∇u6.A further preliminary result consists in the estimates for the Laplace problem (e.g., [15, 16]).Lemma 1.
Supposeu is a weak solution to(15)Δu=finΩ0with eitherau=0orb∂zu=0atz=0,1.
Then, iff∈Wm,2Ω0, m≥0, one has u∈Wm+2,2Ω0 in case (a), and the following inequality holds:(16)um+2,2≤cfm,2,with c independent of u and f. The same result holds in case (b) with u∈Wm+2,2Ω0, provided f∈Wm,2Ω0.In the present framework, once the perturbation fieldsτ=T−Tr and P=p−pr are defined, the following nondimensional system of equations are addressed:(17)∇⋅v=0,1Pr∂v∂t+v⋅∇v=−∇P−βPk+Δv+Raτk,∂τ∂t+v⋅∇τ=Δτ+v⋅k,where Ra and Pr are the Rayleigh and Prandtl numbers, respectively, and k=0,0,1 is directed along the upward vertical direction. Notice that, given p0, the perturbation P has to be zero at the reference state. The boundary conditions are homogeneous for τ, while for v, one imposes the impermeability condition vz=0 and the stress-free condition vzx=vzy=0, so that one could compare the stability results with the classic exact ones obtained under the same conditions. Let us set, just for simplicity, ∇⋅v=ϕ.For “regular” solutions, i.e., solutions, such that(18)v∈L∞0,T;W2,2Ω0∩L20,T;W3,2Ω0,vt∈L∞0,T;L2Ω0∩L20,T;W1,2Ω0,corresponding to v0∈W2,2Ω0 at the initial time, one can show that system (17) is equivalent to(19)ΔP+βPz=−1Pr∇⋅v⋅∇v+ϕv+Raτz,1Pr∂v∂t+v⋅∇v+ϕv=−∇P−βPk+Δv+Raτk,∂τ∂t+v⋅∇τ=Δτ+v⋅k,with Robin’s conditions for P at z=0,1:(20)∇P⋅k+βP=0.The extra nonlinear term lying at the left-hand side of (19)2 is inserted to make the stress-free condition compatible with the equation, by matching the mean value of the horizontal components. As a matter of fact, v⋅∇v+ϕv=∇⋅v⊗v gives rise, once integrated, to a vanishing boundary integral, as any other terms in the horizontal projection of equation (19)2.Actually, in one direction, the implication is obvious: it is sufficient to take the divergence of the second equation in (17) to obtain the first of (19). The boundary conditions for P are established by evaluating at z=0,1 the normal component of (17)2: the trace of vtz and, by regularity, the trace of vzzz are defined and vanished, leading to (20). As a matter of fact, by deriving the divergence-free condition with respect to z, one gets vzxx+vzyy+vzzz=0, and deriving the stress-free condition along the boundary (since one can exchange the order of derivation), one gets vzxx=vxzx=0 and vzyy=vyzy=0, which can be replaced in the previous one.Conversely, let us assume that (19) holds true and the solutions are regular; in particular, assume that P∈L20,T;W2,2Ω0. Let us replace (19)1 in the divergence of (19)2, then one obtains(21)1Pr∂ϕ∂t=Δϕ.Let us remark thatϕ=0. Indeed, if vxx or vyy is different from zero, then their mean value is zero by periodicity. So, one deduces ϕ=vzz=0 because after integration in z∈0,1, one obtains vz. As a consequence, one can use the Poincaré inequality for ϕ.Next, one can perform theL2-inner product of the equation with ϕ: by taking into account the periodicity conditions, the right-hand side of (21) becomes(22)Δϕ,ϕ=12∫01dx∫01dyϕ2zx,y,1−ϕ2zx,y,0−∇ϕ,∇ϕ,from the regularity hypotheses, one can deduce that the boundary terms vanish in the trace sense because of impermeability and stress-free conditions, which implies vzxx=vxzx=0 and vzyy=vyzy=0. On the other hand, Robin’s condition for the pressure implies Δvz=0 in the momentum balance at the boundary. Hence, since one can derive the impermeability condition with respect to x and y, it follows also vzzz=0. Thus, by summing up, one sees that ϕz vanishes at the boundaries.In conclusion, one can write(23)1Prddtϕ22+∇ϕ22≤0.Finally, either the divergence of the velocity is zero att=0 and then for all t>0, or it decays exponentially to zero. This means that the divergence-free condition can be fulfilled and is, moreover, physically observable.Hereafter, a solution with the required regularity is shown to exist, provided the initial data for (19)2 and (19)3 are in W2,2Ω0.Concerning boundary conditions, the change of variableΠ=Peβz turns Robin’s ones into the simpler Neumann conditions:(24)Πz=0,z=0,1,and simplifies the system under study, which becomes like the one proposed in [9], page. 7, and studied in [4]:(25)ΔΠ−βΠz=−eβzPr∇⋅v⋅∇v+ϕv+Raeβzτz,1Pr∂v∂t+v⋅∇v+ϕv−Δv=−∇e−βzΠ−βe−βzΠk+Raτk,∂τ∂t+v⋅∇τ−Δτ=v⋅k.The new variableΠ is defined up to an arbitrary constant, unlike P which appears in the buoyancy force and depends on the boundary value pb which is in the basic solution. Thus, Π can be prescribed with a mean value zero and verify the Poincaré inequality.The trivial solution (v=c1i+c2j, τ=0, P=0) corresponding to the Galilean invariance has to be considered equivalent to the rest state, so for this reason, solutions are looked in Banach subspaces defined by the following condition:(26)vx=vy=0.Next, one assigns inΩ0 the periodicity conditions in the horizontal directions and the stress-free conditions at the boundaries and homogeneous conditions for the deviatory temperature τ:(27)Πzx,y,0,t=Πzx,y,1,t=0,vzx,y,0,t=vzx,y,1,t=0,vzxx,y,0,t=vzxx,y,1,t=0,τx,y,0,t=τx,y,1,t=0.At the initial time, data are given concordant with the previous ones:(28)vx,y,z,0,τx,y,z,0=v0x,y,z,τ0x,y,z.The plan is studying (17) before proving the equivalence with (25). To this end, one looks for solutions such that a.e. in t, and for k∈ℕ0, Π belongs to Wk,2Ω0, which is defined as the closure in the Sobolev space norm of C∞Ω0, whose functions are periodic in x and y, verifies the homogeneous Neumann condition at z=0,1, and has a zero mean value (this last feature is necessary to prove the existence). Analogously, without the isochoric conditions, v≡vx,vy,vz has free components such that vx and vy belong to Wk,2Ω0 (with the mean value zero, as previously noticed), while vz and τ belong to W0k,2Ω0, which is the closure of C0∞Ω0.


## 3. Galerkin Approximation

Still in [9] and in [4], an existence result for Π and further estimates are given in 2D. The proof is unaffected by the dimension, and the results are summarized below.Lemma 2.
If0<β<2π and if f∈L2Ω0, then the problem(29)ΔΠ−βΠz=eβzf,inΩ0,Πzx,y,0=Πzx,y,1=0,forx,y∈ℝ2,with periodic side conditions, has a unique solution Π∈W2,2Ω0 such that Π=0, and the following estimate holds true:(30)Π2,2≤Cβf2.In this lemma and hereafter,Cβ stands for a positive function of β such that limβ⟶0Cβ=c≥0. The same convention is kept for Cβ,Pr,Ra.Furthermore, iff=∇⋅w and w∈L2Ω0 with w⋅k=0 at z=0,1, then in [4], one can also find(31)Π1,2≤Cβw2.If, moreover,f=∇⋅∇⋅u⊗u with u⋅k=0 at z=0,1 and ∇⋅u=0, then in [4], one can also find(32)Π2≤Cβu42.Notice that both these estimates fall into the classic Navier–Stokes estimate:(33)p2≤cv42,as β goes to zero.However, here the case∇⋅v≠0 is, in principle, considered.By taking the data of (29) from system (25), all hypotheses concerning the boundary conditions are verified, and by writing(34)f=−1Pr∇⋅v⋅∇v+ϕv+Ra∇⋅τk,one sees that (31) can be used.Hence,(35)∇Π2≤Cβ1Prv⋅∇v2+ϕv2+Raτ2.Now, let us introduce Galerkin’s “approximate” solutions of (25). Let us denote by φjj∈ℕ and χjj∈ℕ the periodic eigenfunctions of the Laplace operator, respectively, verifying Dirichlet and Neumann conditions at ∂Ω, such that the vector functions Ψj=χj,χj,φj verify the same boundary condition as v does:(36)ΔΨj=−λjΨj,λj>0,Δφj=−ηjφj,ηj>0.Moreover,Ψj’s and φj’s are complete orthogonal bases, respectively, for the vector and scalar fields in W1,2Ω0. Further, they belong to W2,2Ω0. The Galerkin approximation solutions vN and τN,(37)vN=∑j=1NCNjtΨj,τN=∑j=1NANjtφj,are then defined in a standard way by projecting system (25) on the N-dimensional subspace spanned by the same eigenfunctions. The ordinary differential equation system so obtained is the first order in the unknowns CNjt and ANjt. It is an autonomous homogeneous system in a normal form, whose solutions are easily shown to exist for all N∈ℕ.As a matter of fact, although the pressure terms in the momentum balance (which will be written in a compact form ase−βz∇Π) cannot be solved a posteriori as for Navier–Stokes equations, one can nevertheless define for all N∈ℕ some suitable ΠN as a function of CNjt and ANjt. To this end, let us use (34) in Lemma 2 and denote by A the linear operator:(38)AΠN≔ΔΠN−βΠzN=eβz∇⋅−1PrvN⋅∇vN+ϕNvN+RaτNk,where, of course, ϕN=∇⋅vN. By Lemma 2, the operator is continuously invertible for 0<β<2π and(39)ΠN≔−1Pr∑k,i=1NA−1eβz∇⋅Ψk⋅∇Ψi+Ψi∇⋅ΨkCNktCNit+RaA−1∑i=1Neβz∇⋅φikANit,where the summation over repeated indices is understood. According to Lemma 2, the linear operator A−1 is defined in L2Ω0, and its image belongs to W2,2Ω0. Moreover, A−1 does not depend on t.For the sake of brevity, let us denote(40)ℱkix,y,z≔A−1eβz∇⋅Ψk⋅∇Ψi+Ψi∇⋅Ψk,Gix,y,z≔A−1eβz∇⋅φik.Next, let us compute(41)e−βz∇ΠN,Ψj=−1Pr∑k,i=1Ne−βz∇ℱki,ΨjCNkCNi+Ra∑i=1Ne−βz∇Gi,ΨjANi.This expression can be inserted in the Galerkin system, which can be written as(42)C˙Nj=−PrλjCNj−∑k,i=1NΨk⋅∇Ψi−e−βz∇ℱki,ΨjCNkCNi++PrRa∑i=1Nφik−e−βz∇Gi,ΨjANi,(43)A˙Nj=−∑k,i=1NCNkANiΨk⋅∇φi,φj−ηjANj+CNj.It allows for solutions since the right-hand side is a Lipschitz continuous function of the variables (in fact, it is at most quadratic). For allN∈ℕ, the initial conditions CNj0=C0j=v0,Ψj and ANj0=A0j=τ0,Ψj correspond to the initial conditions of (25). Since the aim is still to derive the equivalence of the differential systems (17) and (19) by showing existence with ϕ=0, the required regularity for the initial data is W2,2. In particular, one has to define the space as the closure of linear combinations of the eigenfunctions χj−χj so that vx,vy∈W2,2Ω0, while vz,τ∈W02,2Ω0, which is directly obtained by closing the φj’s.Moreover, an estimate forvtN02 and τtN02 is hereafter needed, for which the left-hand side of (42) evaluated in t=0 has to be increased by directly inserting (34) in estimate (31):(44)e−βz∇ΠN,Ψj≤e−β∇ΠN,Ψj≤Cβ,Pr,RavN⋅∇vN+ϕNvN,Ψj+τN,Ψj≤Cβ,Pr,Ra∑k,i=1NCNkCNiΨk⋅∇Ψi+Ψi∇⋅Ψk,Ψj+ANj.After squaring and summing the inequalities derived by (42) and evaluating at the initial time, one obtains a bound independent of N, estimated by the W2,2-norm of the initial data through Bessel’s inequality. Since the basis is in W2,2Ω0, by linearity(45)ΔvN02=∑j=1Nλjv0,ΨjΨj2=∑j=1Nv0,ΔΨjΨj2=∑j=1NΔv0,ΨjΨj2≤Δv02.Concerning the nonlinear terms, one can compute, for instance,(46)∑j=1N∑k,i=1N∑l,n=1NΨk⋅∇Ψi,ΨjΨl⋅∇Ψn,ΨjCNk0CNi0CNl0CNn0≤∑j=1∞v0⋅∇v0,Ψj2=v0⋅∇v02.Next, these nonlinear terms at the right-hand side can be increased by (12) and (13), through Hölder (p=3/2 and p′=3) and interpolation (p=4/3 and p′=4) inequalities (the procedure is detailed in the proof of Lemma 3). Finally, using the Poincaré inequality and (10), one gets(47)vtN02≤Cβ,Pr,RaΔv02+τ02+Δv022,τtN02≤Cβ,Pr,RaΔτ02+v02+∇v02Δτ02.Here and hereafter, theL2-norms of the two nonlinear terms v⋅∇v and ϕv are estimated exactly in the same way, but for brevity only the first one will be developed.


## 4. Evolutive Estimates and Main Results

Let us formally derive thea priori estimates for solutions of system (25): by quite a standard procedure, they are verified just by Galerkin solutions, denoted here by v,τ, and, as seen, are bound uniformly with respect to N.Lemma 3.
Let us define the energy functions:(48)E1t≔12v22Pr+Raτ22,E2t≔12∇v22Pr+Ra∇τ22.
Then, for sufficiently smallε>0, the following inequalities hold true:(49)dE1dt+c∇v22+Ra∇τ22≤Cβ,Pr,RaE13+E23/2+εΔv22+Cβ,PrRaεv22,(50)dE2dt+cΔv22+RaΔτ22≤Cβ,Pr,RaE23+Cβ,PrRaεE1,(51)dE2dt+cΔv22+Raτ22≤Cβ,Pr,RaE23+Cβ,PrRaεΔv22,where c∈0,1.Proof.
One can test (25) with v,τ, and then one multiplies the third equation by Ra and then sums:(52)ddtE1t+∇v22+Ra∇τ22=−e−βz∇Π,v+2Raτ,vz−1Prv⋅∇v,v−1Prϕv,v−Rav⋅∇τ,τ.
In order to estimate the nonlinear term arising from the coupling with the pressure equation, one uses (35): the nonlinear term at the right-hand side can be increased by (12) and (13), through Hölder (p=3/2 and p′=3) and interpolation (p=4/3 and p′=4) inequalities, so that(53)e−βz∇Π,vτ=0≤cv⋅∇v2v2≤cv3∇v6v2≤cv21/2v61/2Δv2v2≤cv23/2∇v21/2Δv2≤c4ε2v26+cε4∇v22+cε2Δv22.
This term is multiplied byβCβ/Pr, while the linear part of the pressure estimate can be included in the estimate of the buoyancy term and of the energy equation transport term, giving rise (in the whole) to(54)RaCβ,Prτ2v2≤RaCβ,Prε2τ22+12εv22.
Here, the term with the second derivatives does not allow to get an autonomous bound forE1t, so that a further estimate for higher derivatives need to be added. Notice that the convective terms do not vanish, and by also resorting to Young’s inequality, they are bounded as follows:(55)v⋅∇v,v=∫Ω0∇⋅vv22−∫Ω0ϕv22≤12ϕ2v42≤C∇v2v21/2v63/2≤C∇v23,v⋅∇τ,τ=∫Ω0∇⋅vτ22−∫Ω0ϕτ22≤12ϕ2τ42≤C∇v2τ21/2τ63/2≤C∇v2∇τ22≤13C∇v23+23C∇τ23.
Analogously,(56)∫Ω0ϕv2≤∇v23.
By collecting all these estimates, using the Poincaré inequality, increasing∇v23 with E23/2, and absorbing at the left-hand side, where possible, the terms of order ϵ, the first inequality follows.
Now, the balance equations are tested withΔv and Δτ, respectively, and again (35) is inserted.
First, the following equations are written:(57)12Prddt∇v22+Δv22=1Prv⋅∇v+ϕv,Δv+e−βz∇Π,Δv−Raτk,Δv,12ddt∇τ22+Δτ22=v⋅∇τ,Δτ+vz,Δτ,dE2dt+Δv22+RaΔτ22=e−βz∇Π,Δv+Raτ,Δvz+RaΔτ,vz1Prv⋅∇v,Δv+1Prϕv,Δv+Rav⋅∇τ,Δτ.
Again, the pressure-related convective terms are bounded, and then the temperature-depending coupling terms are estimated:(58)e−βz∇Π,Δvτ=0≤cv⋅∇v2Δv2≤v6∇v3Δv2≤cv6∇v21/2D2v21/2Δv2≤c∇v23/2Δv23/2≤3cε4Δv22+c4ε∇v26.
In order to reach local-in-time existence of solutions for any initial data, one writes(59)RaCβ,Prτ2Δv2+Δτ2v2≤RaCβ,Prε2Δτ22+ε2Δv22+12ε2E1t.
In order to achieve global-in-time existence for small initial data, one uses an alternative estimate, which follows by several applications of the Poincaré inequality:(60)RaCβ,Prτ2Δv2+Δτ2v2≤RaCβ,Prε2Δτ22+12εΔv22.
Now, the nonlinear terms are focused:(61)v⋅∇v+ϕv,Δv≤Cv6∇v3Δv2≤Cv6∇v21/2∇v61/2Δv2≤C∇v23/2Δv23/2≤3ε4CΔv22+14εC∇v26,v⋅∇τ,Δτ≤v6∇τ3Δτ2≤C∇v2∇τ21/2D2τ21/2Δτ2≤C∇v2∇τ21/2Δτ23/2≤3ε4CΔτ22+14εC∇τ22∇v24≤3ε4CΔτ22+112εC∇τ26+16εC∇v26.
The terms of orderϵ are still absorbed at the left-hand side, and then inequalities (50) and (51), respectively, follow by (59) and (60).Lemma 4.
Let(62)T≔logE20+1E201/4C,then for all t∈0,T, the following estimates hold:(63)Et≤E01+E201+E0E0−1+E201/2,where Et≔E1t+E2t. Furthermore,(64)∫0TPtdt≤ClogE20+1E201/4E0+E1/20,where Pt≔∇vt22+Δvt22+∇τt22+Δτt22.Proof.
By summing side-by-side the first and second inequalities in Lemma3, one obtains(65)ddtEt+c1Pt≤c2Et+E3/2t+E3t.
Here, one has(66)E3/2=E3/4E3/4≤34E+14E3,and consequently the right-hand side can be bounded by E+E3. Hence,(67)∫E0EtdEEE2+1=∫E0Et1E−EE2+1dE≤Ct,then(68)EtE20+1E0E2t+1≤eCt.
It follows that once the maximal interval corresponding to a vanishing denominator is identified,(69)∀t∈0,Twith2T≔logE20+1E201/2C,(70)E2t≤e2CtE201+E201−e2Ct≤E01+E201+E0E0−1+E20,the energy is bounded.
Next, coming back to (65) and integrating in 0,T, one deduces a bound for the norm in L∞0,T;W1,2Ω0∩L20,T;W2,2Ω0, as can be seen hereafter: the inequality(71)∫0TPtdt≤ClogE20+1E201/4E0+E1/20,implies in particular the integrability of ∇vt22 and of Δvt22.Although, the Galerkin solutions exist for allt>0, their uniform estimates exist only in the finite interval defined in (69). Global-in-time solution will be found only by asking suitable smallness of Ra and of the initial data.Finally, (25) is derived formally with respect to t, and the same procedure is repeated as in the first estimate, but for the nonlinear convective terms. Now, in order to bound the initial data for vt, initial data in W2,2 is chosen:(72)ΔΠt−βΠtz=−eβzPr∇⋅vt⋅∇v+ϕtv+v⋅∇vt+ϕvt+Raeβzτtz,1Pr∂vt∂t+vt⋅∇v+ϕtv+v⋅∇vt+ϕvt−Δvt=−∇e−βzΠt−βe−βzΠtk+Raτtk,∂τt∂t+vt⋅∇τ+v⋅∇τt−Δτt=vtz.Again, the terms depending onϕ or ϕt will be bounded exactly as the corresponding convective ones.Lemma 5.
Let us define the energy function as(73)ℰt≔12vt22Pr+Raτt22.
Then, for sufficiently smallε>0, the following inequality holds true:(74)dℰdtt+c∇vt22+Ra∇τt22≤Cβ,Pr,RaΔv22ℰt+Cβ,PrRaεvt22,where c∈0,1.Proof.
Let us start with the equation(75)ddtℰt+∇vt22+Ra∇τt22=−e−βz∇Πt,vt+2Raτt,vtz+−1Prvt⋅∇v+v⋅∇vt,vt−1Prϕtv+ϕvt,vt−Ravt⋅∇τ+v⋅∇τt,τt.
Technically, the estimates of the nonlinear convective terms are the only different ones from the previous ones, since the terms are not homogeneous:(76)e−βz∇Πt,vtτt=0≤cvt2vt⋅∇v2+v⋅∇vt2≤cvt2vt3∇v6+vt2v∞∇vt2≤cvt23/2∇vt21/2Δv2+vt2Δv2∇vt2≤cΔv22vt22+ε∇vt22.
The same estimates can be repeated for the other nonlinear terms, such as(77)ϕvt,vt≤c∇v6vt2vt3,ϕtv,vt≤cvt2v∞∇vt2,vt⋅∇τ,τt≤c∇τ6vt2τt3,v⋅∇τt,τt≤cτt2v∞∇τt2.
The gradient terms can be absorbed at the left-hand side.Starting from (74), where the positive term at the left-hand side can be neglected and the term vt22 at the right-hand side can be bounded by ℰt, by a generalized Gronwall argument, one shows that(78)ℰt≤ec∫0TΔvt22+1dtℰ0.As a matter of fact, (71) implies v∈L20,T;W2,2Ω0, so that ℰt is bounded in 0,T. Further, by coming back to (74) and integrating again, it follows also(79)∫0T∇vt22+∇τt22dt≤Ov02,2,τ02,2,and this is independent of the size of Ra.Theorem 1.
Letβ<2π and Pr and Ra be arbitrary, then system (25) with initial data v0,τ0∈W2,2Ω0 admits a solution a la Ladyzhenskaya, namely,(80)v,τ∈L∞0,T;W2,2Ω0∩L20,T;W3,2Ω0,vt,τt∈L∞0,T;L2Ω0∩L20,T;W1,2Ω0,where(81)T≔logE20+1E201/4C.
Moreover, if∇⋅v0=0, then ∇⋅vt=0 for all t∈0,T. Finally, if Ra, v02,2, and τ02,2 are sufficiently small, depending on Pr and β, then T=∞ is taken in (80) and v,τ decays exponentially fast to zero.Proof.
Employing the estimates proved in Lemma4, it is routine to show that the approximate solutions constructed with the Galerkin method converge to a solution:(82)v,τ∈L∞0,T;W1,2Ω0∩L20,T;W2,2Ω0,Π∈L20,T;W2,2Ω0,vt,τt∈L∞0,T;L2Ω0∩L20,T;W1,2Ω0,where T is given in (81).
The 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 vt2, as well as the integrability of ∇vtt22. To this end, viewing the momentum balance as a Poisson problem and increasing the pressure in the usual way, the following is obtained with the help of Lemma 1:(83)vt2,2≤Cv⋅∇v2+∇Π2+vt2+τ2≤C∇v24+εΔv22+vt2+τ2.
Theε term at the right-hand side can be absorbed by the left-hand side, so that finally the following bound is obtained:(84)esssup0,Tvt2,2≤Cesssup0,T∇vt24+τ2+vt2≤Ov02,2,τ02,2.
Moreover, again from Lemma1, it is also obtained that(85)∫0Tvt3,22dt≤C∫0Tv⋅∇v1,22+Π2,22+vt1,22+τ1,22dt.
It is remarked that, as noticed in Section2, the compatibility condition related to the Neumann problem is verified. Here, the nonlinear terms at the right-hand side can be increased by using (14) and interpolation inequality:(86)v⋅D2v2≤cv∞D2v2≤cD2v22,∇v∇vT≤c∇v42≤c∇v6∇v3≤cD2v22.
The proof thatτ is bounded in L∞0,T;W1,2Ω0∩L20,T;W2,2Ω0 is similar and simpler.
The proof of global existence follows from Lemmas3 and 5 by setting(87)E0t≔Et+ℰt,P0t≔∇vt22+Δvt22+∇vtt22+Ra∇τt22+Δτt22+∇τtt22.
In fact, if inequalities (49), (51), and (74) are added, bringing to the left-hand side the three terms with factor Ra/ε and using the Poincaré inequality(88)E≤kP,to absorb them in P0 for sufficiently small Ra (depending on ε), one gets(89)ddtEt+c1Pt≤c2E3/2t+E3t.
The constantc1 is now fixed positive by suitably restricting Ra. Again, using the Poincaré inequality on the left-hand side of (89), one infers(90)ddtEt+c1−c2E1/2t+E2t1kEt≤0.
Thus, the exponential decay follows by choosing at the initial time:(91)E1/20+E20≤12c1c2.
In fact, by continuity, for some time,Et≤E0. But, this reinforces the previous condition so that the positive function in square brackets cannot decrease and is bounded from below by a positive constant for all times.



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*Source: 9610689-2020-07-08.xml*