On the Incompressible Limit of a Strongly Stratified Heat Conducting Fluid

A compressible, viscous and heat conducting fluid is confined between two parallel plates maintained at a constant temperature and subject to a strong stratification due to the gravitational force. We consider the asymptotic limit, where the Mach number and the Froude number are of the same order proportional to a small parameter. We show the limit problem can be identified with Majda’s model of layered “stack-of-pancake” flow.


Introduction
Consider the motion of a compressible viscous and heat conducting fluid confined between two parallel plates. For simplicity, we suppose the motion is space-periodic with respect to the horizontal variable. Consequently, the spatial domain Ω may be identified with Ω = T 2 × (0, 1), T 2 = [−1, 1] The time evolution of the fluid mass density = (t, x), the absolute temperature ϑ = ϑ(t, x), and the velocity u = u(t, x) is governed by the Navier-Stokes-Fourier (NSF) system:

Asymptotic Limit
In accordance with the scaling of (1.2), (1.3), the zero-th order terms in the asymptotic limit are determined by the stationary (static) problem (1.9) Applying curl operator to identity (1.9), we successively deduce where we have anticipated that the pressure also depends non-trivially on the temperature ϑ and is such that ∂p( ,ϑ) ∂ϑ = 0. Thus for the static problem (1.9) to be solvable, both ∇ x and ∇ x ϑ must be parallel to ∇ x G. This fact imposes certain restrictions on the distribution of the boundary temperature ϑ B . In particular, the motion in an inclined layer studied by Daniels et al. [5] does not admit any static solution. Accordingly, we focus on the particular case (1.10) Fixing the temperature profile ϑ B = Θ(x 3 ) to comply with the boundary conditions (1.10), we may recover = r(x 3 ) as a solution of the ODE Needless to say, such a problem may admit infinitely many solutions.
To simplify, we focus on the case Θ bott = Θ up > 0. Accordingly, we consider the reference temperature profile Θ = Θ bott = Θ up -a positive constant. Then it follows from (1.11) that the static density profile r = r(x 3 ) must be non-constant as long as g = 0. Anticipating the asymptotic limit ε → r, ϑ ε → Θ, u ε → U (in some sense) we deduce from the equation of continuity (1.1) div x (rU) = 0. (1.12) As r depends only on the vertical x 3 -variable, this yields (1.14) In view of the previous arguments, the limit fluid motion exhibits the "stack of pancakes structure" described in Chapter 6 of Majda's book [14]. Specifically, U = [U h , 0], and Here and hereafter, the subscript h refers to the horizontal variable The fluid motion is purely horizontal, the coupling between different layers only through the vertical component of the viscous stress.
To the best of our knowledge, there is no rigorous justification of the system (1.15)-(1.17) available in the literature except the inviscid case discussed in [7]. It is worth noting that a similar problem for the barotropic Navier-Stokes system gives rise to a different limit, namely the so-called anelastic approximation, see Masmoudi [15] or Feireisl et al. [8]. Furthermore, as observed in [3], the related case of a low stratification with Ma = ε and Fr = √ ε leads to a limiting system of Oberbeck-Boussinesq type with non-local boundary conditions for the temperature.

The Strategy of the Convergence Proof
We start with the concept of weak solutions for the NSF system with Dirichlet boundary conditions introduced in [4]. In particular, we recall the ballistic energy and the associated relative energy inequality in Sect. 2. Next, we introduce the concept of strong solutions to Majda's system in Sect. 3. In Sect. 4, we state our main result.
The strategy is of type "weak" → "strong", meaning the strong solution of the target system is used as a "test function" in the relative energy inequality associated to the primitive system. In Sect. 5, we derive the basic energy estimates that control the amplitude of the fluid velocity as well as the distance of the density and temperature profiles from their limit values independent of the scaling parameter ε. In Sect. 6, we show convergence to the target system (1.15)-(1.17) anticipating the latter admits a regular solution. This formal argument is made rigorous in Sect. 7, where global existence for Majda's model is established. The last result may be of independent interest.

Relative Energy Inequality
In addition to Gibbs' equation (1.8), we impose the hypothesis of thermodynamic stability written in the form Next, following [4], we introduce the scaled relative energy Now, the hypothesis of thermodynamic stability (2.6) can be equivalently rephrased as (strict) convexity of the total energy expressed with respect to the conservative entropy variables , whereas the relative energy can be written as Finally, as observed in [4], any weak solution in the sense of Definition 2.1 satisfies the relative energy inequality for a.a. τ > 0 and any trio of continuously differentiable functions (˜ ,θ,ũ) satisfying

Constitutive Relations
The existence theory developed in [4] is conditioned by certain restrictions imposed on the constitutive relations (state equations) similar to those introduced in the monograph [9, Chapters 1,2]. Specifically, the equation of state reads where p m is the pressure of a general monoatomic gas, Moreover, using several physical principles it was shown in [9, Chapter 1]: • Gibbs' relation together with (2.9) yield • Hypothesis of thermodynamic stability (2.6) expressed in terms of P gives rise to In particular, the function Z → P (Z)/Z 5 3 is decreasing, and we suppose • Accordingly, the associated entropy takes the form In addition, we impose the Third law of thermodynamics, cf. Belgiorno [1,2], requiring the entropy to vanish when the absolute temperature approaches zero, Finally, we suppose the transport coefficients are continuously differentiable functions satisfying As a consequence of the above hypotheses, we get the following estimates:

Strong Solutions to Majda's System
Problem (1.16)-(1.17) shares many common features with the 2d−incompressible Navier-Stokes system solved in the celebrated work by Ladyženskaja [12,13]. Indeed, we show that problem (1.16)-(1.17), endowed with the boundary conditions is globally well posed in the framework of Sobolev spaces W 2,p with p > 1 large enough. We report the following result that may be of independent interest.
Let the initial data U 0,h belong to the class for all 1 ≤ q < ∞.

Remark 3.2.
To avoid any misunderstanding we emphasize that by Similarly, The proof of Theorem 3.1 is postponed to Sect. 7.

Main Result
Having collected the necessary preliminary material, we are ready to state our main result. and let Let ( ε , ϑ ε , u ε ) ε>0 be a family of weak solutions of the scaled NSF system in the sense of Definition 2.1 emanating from the initial data and U 0,h belongs to the class (3.3). Then

4)
where U h is the unique solution of Majda's system, the existence of which is guaranteed by Theorem 3.1.
Hypothesis (4.3) corresponds to well-prepared initial data. In view of the coercivity properties of the relative energy stated in (5.1), (5.2) below, relation (4.4) implies, in particular, The next two sections are devoted to the proof of Theorem 4.1.

Uniform Bounds
In order to perform the singular limit in the NSF system we need the associated sequence of weak solutions ( ε , ϑ ε , u ε ) ε>0 to be bounded at least in the energy space. First, we introduce the notation of [9] to distinguish between the "essential" and "residual" range of the thermostatic variables ( , ϑ). Specifically, given a compact set As shown in [9, Chapter 5, Lemma 5.1], the relative energy enjoys the following coercivity properties:

Conclusion, Uniform Bounds for Ill-Prepared Data
In view of the estimates obtained in the previous section, we deduce from (5.7) for ill-prepared initial data satisfying (5.4) the following bounds independent of the scaling parameter ε → 0: Next, it follows from (5.8) that the measure of the residual set shrinks to zero, specifically In addition, we get from (5.8): ess sup Combining (5.10), (5.11), and (5.12), we conclude Finally, we claim the bound on the entropy flux Indeed we have where the former term on the right-hand side is controlled via (5.13). As for the latter, we deduce from (5.10) that hence it is enough to check To see (5.15) first observe that ess sup and, in view of (5.10) and Poincaré inequality, Consequently, (5.15) follows by interpolation. Of course, the above uniform bound remain valid also for the well-prepared initial data considered in Theorem 4.1.

Convergence to the Target System
We show convergence to the regular solution U h in Majda's system claimed in Theorem 4.1. To get a lean notation, we will identify the two-dimensional velocity U h with its three-dimensional counterpart [U h , 0]. The ansatz (˜ ,θ,ũ) = (r, Θ, U h ) in the relative energy inequality (2.7) yields where we have used the stationary equation Next, seeing that Now, in view of the uniform bounds (5.9), (5.12), where Q(ε) denotes a generic function with the property Q(ε) → 0 as ε → 0. Next, in view of (5.9), (5.12), we may assume up to a suitable subsequence, where div x (ru) = 0. As entropy is given by the constitutive equation (2.13), (2.14), ∂s(r, Θ) ∂ < 0, and we conclude In addition, since U h , u satisfy (1.17), (6.5), respectively, we obtain Going back to (6.2), we deduce Finally, exploiting weak lower semi-continuity of convex functions, we conclude which, applying the standard Grönwall argument, yields the desired convergence as well as u = U h . We have proved Theorem 4.1.

Global Existence for Majda's Problem
Our ultimate goal is to show global existence of strong solutions to Majda's model claimed in Theorem 3.1. To this end, it is more convenient to consider the (horizontal) vorticity formulation of (1.16), (1.17).
With a slight abuse of notation in the definition of U h , this formulation reads with the boundary conditions ω| ∂Ω = 0, (7.4) and the initial condition For given ω, the velocity field U h can be recovered via Biot-Savart law: Remark 7.1. Strictly speaking, the velocity U h is determined by (7.7) up to its horizontal average that can be recovered as the unique solution of the parabolic problem

Construction via a Fixed Point Argument
The desired solution ω to (7.1)-(7.5) can be constructed via a simple fixed point argument. Consider the set As the initial velocity U 0,h belongs to the class (3.3), the set X M is a bounded closed convex subset of the Banach space C([0, T ] × Ω). Moreover, X M is non-empty as long as M is large enough to accommodate the initial condition.
Since this bound is independent of L, we may choose L large enough so that b L (U h ) = U h to get the desired conclusion Finally, it is easy to check that the solution is unique in the regularity class (3.4). As a matter of fact, a more general weak-strong uniqueness holds that could be shown adapting the above arguments based on the relative energy inequality.
We have proved Theorem 3.1.

Funding Information
Open access publishing supported by the National Technical Library in Prague.

Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Open Access. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.