Existence and Uniqueness of Isothermal, Slightly Compressible Stratified Flow

We show well-posedness for the equations describing a new model of slightly compressible fluids. This model was recently rigorously derived in Grandi and Passerini (Geophys Astrophys Fluid Dyn, 2020) from the full set of balance laws and falls in the category of anelastic Navier–Stokes fluids. In particular, we prove existence and uniqueness of global regular solutions in the two-dimensional case for initial data of arbitrary “size”, and for “small” data in three dimensions. We also show global stability of the rest state in the class of weak solutions.


Introduction
The Oberbeck-Boussinesq approximation [1,20] is a very popular model, used to describe convection in a horizontal layer of fluid heated from below [14]. As is well known, the basic assumption is that the fluid is incompressible, namely, the velocity field v satisfies ∇ · v = 0 at all points and times, whereas the density depends solely on the temperature T , and its contribution becomes relevant only in the buoyancy term.
However, as noticed for instance in [10,11,18,21], thermodynamic variables such as energy and density cannot be a function of T only, but should also depend on the pressure p since, otherwise, Gibbs law would be unattended and stability in waves propagation not allowed. It turns out that, for these more general models, one has necessarily to relax the solenoidal condition on v [6,17] to enlarge the region of the non-dimensional parameter space in which formal limits lead to reliable approximations for compressible fluids.
Motivated by this important issues, the author jointly with D. Grandi rigorously derived, by perturbative methods from the full set of balance laws, new models where the density, ρ, may depend on both T and p [12,13], while for related ones, the authors addressed well-posedness and stability questions, in [2,4,5]. In particular, in [13], we proposed a new model for thermal convection in a horizontal layer of fluid heated from below with ρ = ρ(T, p). In such a case, the original full compressible system allows for an elementary solution s e := (T e , p e ) with corresponding ρ e = ρ(T e , p e ); see (2.2), (2.4), and (2.5). The relevant equation are then obtained from the full system by a perturbation expansions around s e , by using as non-dimensional small parameter αδT , where α is the thermal expansion coefficient and δT is the temperature difference between the two horizontal planes confining the fluid. The limiting equations thus derived (see (OB) β ) fall, in the isothermal case, in the category of the so called anelastic approximations of the Navier-Stokes equations; see (2.9), where the velocity field is no longer solenoidal but, instead, satisfies ∇ · (ρv) = 0. It is just the latter that makes the new model interesting from the point of view of well-posedness of the corresponding initial-boundary value problem, which constitutes the focus of this article.

Formulation of the Problem
Assume the fluid occupies a horizontal layer comprised between the unmovable planes placed at z = 0 and z = h, subject to the gravity force ρg. The planes are kept at constant, not necessarily equal, temperatures namely, T = T d at z = 0 and T = T d − δT , δT ∈ R, at z = h. Under isothermal conditions δT = 0, the constitutive equation ρ = ρ 0 = constant allows for the fluid to be at rest (v ≡ 0) with corresponding hydrostatic pressure p = −ρgz. On the other hand, if δT = 0, then the fluid is still at rest while the "classical" Boussinesq constitutive equation implies the well-known linear profile for the temperature The latter, in turn, furnishes the following expression for the pressure field [1] In [18], the author jointly with T. Ruggeri have introduced a constitutive equation more general than (2.1) to include also pressure variations: In such a case, while the stratified temperature field remains unchanged, the pressure distribution becomes [18]: Notice that in the isothermal case δT = 0, combining (2.4) and (2.3), we find ρ e = ρ d e −βρ d gz , (2.5) which provides the compatible stratification of the fluid.
The main accomplishment of [13] was to derive, as formal limit from the full compressible model, an approximate set of equations for the perturbation (τ, P, v) to the basic solution (T e , p e , v ≡ 0), that generalizes the classical O-B system by taking into account the compressibility of the fluid (anelastic O-B system). More precisely, setting in [13] it is shown that (τ, P, v) must satisfy the following nondimensional equations 1 where τ, P, v and β are meant now to be nondimensional quantities. Moreover, with μ, ζ and κ being, respectively, shear and bulk viscosities and thermal conductivity of the fluid. Finally, ξ = ±1, according to whether δT ≶ 0.

Statement of the Main Results
Our main goal is the investigation of the well posedeness of the initial-boundary value problem associated to (2.9). As in the case of the classical Navier-Stokes model, the result may depend on the Euclidean dimension. We begin to give the definition of weak solution.
We now collect the main results proved in this paper in the form of as many theorems. We begin with the following one. Theorem 3.1. Let n = 2, 3 and let Pr, γ and β be given. Then, for any initial data v 0 ∈L 2 (Ω 0 ), there exists at least one weak solution for all T > 0. Moreover, such a solution satisfies the following decay We also prove existence and uniqueness of more regular solutions, in the form stated in the following theorems.

Theorem 3.2.
Let Ω 0 ⊂ R 2 . Then, given arbitrary positive Pr, γ and β, and arbitrary initial data in W 1,2 (Ω 0 ), there exists a unique corresponding weak solution v to (2.7) which, in addition, is in the  Ω 0 × (0, T ). Finally, there exists a constant C 0 depending only on the above physical parameters, such that if v 0 2 2 ∇v 0 2 2 ≤ 2/C 0 we can take T = ∞. Remark 3.1. The proofs of existence are based on the Galerkin method, in the way suggested by Prodi [19]. However, Prodi's approach must be suitably modified, in that, in our case, the velocity field is no longer solenoidal. This requires the study of full regularity of solutions to a new Stokes problem derived in connection with problem (2.9); see (4.11). The latter, in turn, requires the study of an appropriate Helmholtz-like decomposition of the type L 2 (Ω 0 ) = L 2 (Ω 0 )⊕G β (Ω 0 ), dictated by the the non-solenoidality of the velocity field (see (4.9)). All these results are contained in Sect. 3, which, in fact, constitutes the heart of the matter of the paper.

Remark 3.2.
In the case of three-dimensional weak solutions (Theorem 3.1), following [3] and [16], one can proves a result of strong convergence of the Galerkin approximation (the first known for weak solutions) in L q (0, T ; W 1,2 (Ω 0 )) for all q ∈ [1,2). Actually, in [3] it is stated that such a convergence occurs along the Galerkin approximation with aspecial base. For the sake of the brevity we do not give the details. The interest of this convergence consists in the fact that one can show a sort of energy equality, or equivalently, an evaluation of the possible gap in the inequality see [3].

Preliminary Results
We start by proving some formal properties of solutions to (2.9)-(2.10). To this end, we observe that, in view of (2.9) 1 , the vector field u := e −βz v is solenoidal, in fact As a consequence, since the factor e −βz can be associated to the first term of the bilinear form as well, by integrating by parts and using (2.10) and (4.1), one shows that the norms v 2,β and v 2 are equivalent. Property (4.3) will be used throughout, even without explicitly mentioning it. Now, if we formally dot-multiply both sides of (2.9) 1 by the solenoidal field u, integrate by parts over Ω 0 and use (4.2) and (2.10), we deduce the following important relation, for all β ≥ 0 We also notice that, using (4.1) and assuming, without loss that the (constant) equal temperatures on the bounding planes are 0, (formally) multiplying both sides of (2.8) by τ and integrating over Ω 0 , we get 1 2 Employing on the right hand side of this equation the scalar version of (4.25) in conjunction with Gronwall's lemma, we obtain, as expected, an exponential decay to the boundary temperature. Now, we derive some results that will play an important role for the existence results developed in the next section. We begin to prove the unique solvability of the following Neumann problem: in a suitable function class. In order to reach this goal, we observe that (4.5) is (formally) equivalent to the following one In view of Wirtinger inequality, H becomes a Hilbert space with respect to the norm induced by the scalar product We shall say that q ∈ H is a weak solution to (4.6) if Let us denote by H −1 the dual space of H. We show the following lemma that in the case α < 2π was proved in [2,4] by different arguments.
Proof. Existence of a unique weak solution q is an immediate consequence of the assumption on G and of Riesz representation theorem. Moreover, setting φ ≡ q in (4.7) we deduce the existence of a positive constant C such that Next, assume G ∈ L 2 (Ω 0 ) and set ψ := e −αz ϕ. From (4.7) we thus get Since F ∈ L 2 (Ω 0 ), by classical elliptic regularity we get that q ∈ W 2,2 (Ω 0 ), and that it satisfies (4.5) along with q 2,2 ≤ C M 2 . The latter, combined with (4.8) completes the proof of the lemma.
An important consequence of the previous result is given in the following lemma.
The next result is a corollary to Lemma 4.3.

23)
Proof. The previous lemma shows that the operator A is surjective with a compact inverse. Since A is symmetric, this implies that A is selfadjoint with a purely discrete spectrum. The lemma is then a consequence of classical results.
We also have to generalize the Friedrichs inequality as follows.
We end this section by recalling a number of classical inequalities that will be frequently employed and valid for a vector function v : Ω 0 → R n , n = 2, 3, satisfying (2.10). First of all, the Poincaré inequality where γ P is a (positive) numerical constant. In what follows, we denote by the same symbol C different constants depending, at most, on the domain Ω 0 . Integrating by parts over Ω 0 one gets ∇v 2 2 = −(Δv, v), and so, by Poincaré and Schwarz inequalities, we deduce We recall Ladyzhenskaya's inequality (see [9]): (4.31) Combining the latter with Poincaré inequality, we infer, in particular, v 4 ≤ C ∇v 2 , n = 2, 3.

Proof of Theorem 3.1
We shall employ Galerkin method with the special basis {Ψ j } introduced in Lemma 4.4. For each N ∈ N, we look for an "approximate solution" defined by where the coefficients C N j (t) are searched as solutions to the the following system of ODE's: Since the matrix B ij is symmetric and, in view of (4.3), positive definite, the system of differential equations (5.1) can be put in normal form. Notice that (5.1) can be equivalently rewritten as Because the involved nonlinear terms are quadratic, it follows that for all N ∈ N, (5.1) has one and only one solution C N := (C N 1 , . . . , C N N ) in some time interval (0, T N ). Clearly, T N = ∞ if we can show that |C N (t)| is uniformly bounded. To this end, we multiply each side of the equation in (5.2) by C N j , sum over j and argue as in (4.1) to get (since the basis function are regular) Integrating both sides of (5.3) from 0 to T N and observing that v 0 N 2,β ≤ C v 0 2 , it follows at once the desired property for C N . Moreover, also with the help of (4.25), we also infer that the sequence {v N } satisfies the following estimate In particular, the latter implies the existence of v ∈ L ∞ (0, T ;L 2 (Ω 0 )) ∩ L 2 (0, T ; W 1,2 (Ω 0 )), all T > 0, (5.5) and of a subsequence It is easy to show that the latter, combined with (5.3), (4.25) and Gronwall's lemma, leads to (3.2). Next, by following more or less classical arguments in conjunction with Lemma 4.5, we shall show that v N k → v strongly in L 2 (0, T ; L 2 (Ω 0 )) .
Actually, from (5.2) and (4.30) we easily get with arbitrary t 1 and t 2 , and Φ j := e −βz Ψ j From (5.8), by Cantor's diagonalization process one can show by standard methods (e.g. [8]) that If we now apply (4.24) to v N k − v, integrate both sides of the resulting equation from 0 to T and use (5.3), we get Thus, letting N k → ∞ in this relation and employing (5.9), by the arbitrariness of ε we arrive at (5.7). From now on the procedure to prove that v is in fact the weak solution is fully standard: we first observe that from (5.2) it follows that (5.10) If we let N k → ∞ in (5.10) we can show, in view of (5.6), that the limit function v satisfies (3.1) with Ψ ≡ Ψ j , provided we also prove 11) or, equivalently, In view of (5.6), it follows that the second term on the left hand side goes to zero. Concerning the fisrst one, we notice that by Hölder and Sobolev inequalities, we deduce Next, from (4.31) 1 and (4.31) 2 we infer that the right hand side in this inequality can be increased by 3 4 2 ∇v N k 2 , n = 3.

Proofs of Theorems 3.2 and 3.3
In this section we shall show that, provided the initial data are more regular, the corresponding Galerkin approximation v N belongs, uniformly in N , to a better regularity class (the so-called "Prodi class"). This will be achieved through suitable "energy estimates" that will eventually lead to the proofs of Theorems 3.2 and 3.3. In this regard, we need some preliminary results. By multiplying both sides of (5.2) byĊ N j , sum over j from 1 to N and integrate by parts as necessary to infer Likewise, multiplying both sides of (5.2) by −λ (j) C N j , summing over j from 1 to N the resulting equation Our next task is to give a suitable estimate of the terms on the right hand side of (6.1) and (6.2). To this end, we observe that, by Lemma 4.3, we have with C independent of N .
Proof of Theorem 3.3. We proceed in a slightly different way with respect to the previous two-dimensional case. We consider (6.6) and increase the right hand side by means of (6.4) and (6.9), choosing suitably i , i = 1, 2, 3, we deduce the following differential inequality in D N (t) := ∇v N (t) 2 2 : dD N dt + κ 1 Δv N (t) 2 2 + κ 2 (v N ) t (t) 2 2 ≤ C 1 D 3 N , for all N and t > 0 . (6.16) where the quantities κ i , i = 1, 2, and C 1 depend only on the physical parameters and i , i = 1, 2, 3. We integrate the differential inequality (6.16) two times. The former concerns simply a bound for D(t). This bound is on some (0, T ) for arbitrary data. Instead the bound is global, that is (0, ∞) for initial data small in the sense indicated in the statement. Without considering the "dissipative terms", by integrating we obtain where we set T := 1/2C 0 ∇v(0) 2 2 . In the case of small data we modify the integration as follows: where we have taken into account (5.4) of the energy inequality. As a result, we have two uniform bounds (one local and the other global in time) for the sequence { ∇v N (t) }. Hence, integrating both sides of (6.16) we get, in particular, where T is finite or infinite depending on the "size" data. Proceeding exactly as in the case n = 2, one can show that the approximating solutions {v N } satisfies (6.11) and (6.12), thus implying that the limit field v is in the class (6.13).