1 Introduction

There are numerous examples of turbulent fluid motion excited by changes of the boundary temperature, among which is the well studied problem of Rayleigh–Bénard convection, see e.g. Davidson [8]. Motivated by similar problems in astrophysics of gaseous stars, we consider a general compressible viscous possibly rotating fluid, occupying a bounded domain \(\Omega \subset R^d\), \(d=2,3\), driven by periodic changes of boundary temperature. The relevant system of field equations for the standard variables: the mass density \(\varrho = \varrho (t,x)\), the velocity \(\textbf{u}= \textbf{u}(t,x)\), and the (absolute) temperature \(\vartheta = \vartheta (t,x)\) reads:

figure a

where \({\mathbb {S}}\) is the viscous stress given by Newton’s rheological law

$$\begin{aligned} {\mathbb {S}} (\vartheta , {\mathbb {D}}_x\textbf{u}) = \mu (\vartheta ) \left( \nabla _x\textbf{u}+ \nabla _x^t \textbf{u}- \frac{2}{3} {\textrm{div}}_x\textbf{u}{\mathbb {I}} \right) + \eta (\vartheta ) {\textrm{div}}_x\textbf{u}{\mathbb {I}}, \end{aligned}$$
(1.4)

and \(\textbf{q}\) is the heat flux given by Fourier’s law

$$\begin{aligned} \textbf{q}(\vartheta , \nabla _x\vartheta ) = - \kappa (\vartheta ) \nabla _x\vartheta . \end{aligned}$$
(1.5)

The momentum equation is augmented by the Coriolis force with the rotation constant vector \(\varvec{\omega }\), the associated centrifugal force as well as the gravitation and other possible inertial time-periodic forces are regrouped in the potential G. The fluid occupies a bounded smooth domain \(\Omega \subset R^d\), \(d=2,3\) endowed with the Dirichlet boundary conditions

figure b

The functions \(\vartheta _B=\vartheta _B(t,x)\) and \(G = G(t,x)\) are smooth and T-periodic in the time variable,

figure c

Hereafter, the problem (1.1)–(1.6) is referred to as Navier–Stokes–Fourier system.

Our goal is to show the existence of a time–periodic solution to problem (1.1)–(1.7). There is a substantial number of references, where such a result is proven under some smallness and smoothness assumption on the data. Valli and Zajaczkowski [24, 25] observe that the distance of two smooth global in time solutions decays in time for the system close to a stable equilibrium, and, as a by product, they deduce the existence of a time periodic solution. Similar ideas have been followed by many authors, see Březina and Kagei [4, 5], Jin and Yang [17] , Kagei and Oomachi [18], Kagei and Tsuda [19], Tsuda [23] to name only a few. Turbulent fluid flows given by large forces out of equilibrium are mostly considered in the framework of weak solutions. Based on the mathematical theory of compressible fluids developed by Lions [20, 21], the existence of large time periodic solutions for the simplified isentropic system was proved in [9] for the isentropic pressure–density equation of state \(p(\varrho ) = a \varrho ^\gamma \), \(\gamma \ge \frac{9}{5}\). The later development of the theory in [12] enabled to extend the result to the case \(\gamma > \frac{5}{3}\), see Cai and Tan [6].

The situation is more delicate for the complete fluid systems including thermal effects. As a direct consequence of the Second law of thermodynamics, the existence of (forced) time periodic solutions is ruled out for problems with purely conservative boundary conditions, see [13]. In [10], the heat flux was controlled by means of a Robin type boundary condition

$$\begin{aligned} \textbf{q} \cdot \textbf{n} = d (\vartheta - \Theta _0) \ \text{ on }\ \partial \Omega , \end{aligned}$$
(1.8)

with a given “mean” temperature \(\Theta _0\). Accordingly, the internal energy is transferred out of the fluid domain in the high temperature regime and the time periodic motion is possible, see [10, Theorem 1]. Our goal is to show a similar result for the Dirichlet boundary conditions (1.6). Note that the problem is much more delicate than in [10] as the heat flux through the boundary is a priori not controlled. Additional novelty is that the function \(\vartheta _B\) in (3.1) is time dependent whereas its counterpart \(\Theta _0\) in [10], cf. (1.8), depends only on x. Finally we note that the presence of the Coriolis force in the momentum equation, though physically relevant in some situations, does not represent any extra analytical difficulties.

Our approach is based on several rather new ideas that appeared only recently in the mathematical theory of open fluid systems.

  • The concept of weak solution for the Navier–Stokes–Fourier system based on a combination of the entropy inequality and the ballistic energy balance developed in [7].

  • Uniform bounds and large time asymptotics of the weak solutions in the spirit of [14].

  • An approximation scheme based on a penalization of the Dirichlet boundary conditions via (1.8).

The concept of weak solution developed in the monograph [11] and used in [10] is based on the total energy balance as an integral part of the definition of weak solution to the Navier–Stokes–Fourier system. This approach applies solely to problems with conservative boundary conditions, where the energy flux vanishes on the boundary of the physical space or it is at least controlled as in (1.8). The problems with inhomogeneous Dirichlet boundary conditions require an alternative approach developed in [7], where the energy is replaced by the ballistic energy, for which the boundary flux is again controllable. This approach has been used recently in [14], where the existence of bounded absorbing sets and asymptotic compactness of bounded trajectories was established.

The constitutive restrictions imposed on the equations of state as well as the transport coefficients are the same as in the existence theory [7]. In particular, the general equation of state of real monoatomic gases proposed in [11, Chapters 1,2] is included. From this point of view, the result is apparently better than in the isentropic case studied in [9], and later revisited by Cai and Tan [6], where the condition \(\gamma > \frac{5}{3}\) is needed. The price to pay is the potential form of the driving force \(\textbf{f} = \nabla _xG\) that, however, includes the physically relevant centrifugal as well as gravitational forces.

The paper is organized as follows. In Sect. 2, we introduce the basic hypotheses concerning the constitutive relations and state the main result. In Sect. 3, we introduce an approximation scheme inspired by [10]. Section 4 is the heart of the paper. Here we establish the necessary uniform bounds to perform the limit in the sequence of approximate solutions. Finally, in Sect. 5, we obtain the desired solution as a limit of the approximate sequence.

2 Main result

Before stating the main result, we recall the form of the constitutive equations proposed in [11, Chapters 1,2]. To comply with the Second law of thermodynamics, we postulate the existence of entropy s, related to the internal energy e and the pressure p through Gibbs’ equation

$$\begin{aligned} \vartheta D s(\varrho , \vartheta ) = D e(\varrho , \vartheta ) + p(\varrho , \vartheta ) D \left( \frac{1}{\varrho } \right) . \end{aligned}$$
(2.1)

2.1 Constitutive theory

Similarly to [11, Chapters 1,2] we consider the pressure equation of state in the form

$$\begin{aligned} p(\varrho , \vartheta ) = p_{\textrm{m}} (\varrho , \vartheta ) + p_{\textrm{rad}}(\vartheta ), \end{aligned}$$

where \(p_{\textrm{m}}\) is the pressure of a general monoatomic gas related to the internal energy through

$$\begin{aligned} p_{\textrm{m}} (\varrho , \vartheta ) = \frac{2}{3} \varrho e_{\textrm{m}}(\varrho , \vartheta ), \end{aligned}$$
(2.2)

augmented by the radiation pressure

$$\begin{aligned} p_{\textrm{rad}}(\vartheta ) = \frac{a}{3} \vartheta ^4,\ a > 0. \end{aligned}$$

Similarly, the internal energy reads

$$\begin{aligned} e(\varrho , \vartheta ) = e_{\textrm{m}}(\varrho , \vartheta ) + e_{\textrm{rad}}(\varrho , \vartheta ),\ e_{\textrm{rad}}(\varrho , \vartheta ) = \frac{a}{\varrho } \vartheta ^4. \end{aligned}$$

Now, Gibbs’ equ. (2.1) gives rise to a specific form of \(p_m\),

$$\begin{aligned} p_m (\varrho , \vartheta ) = \vartheta ^{\frac{5}{2}} P \left( \frac{\varrho }{\vartheta ^{\frac{3}{2}} } \right) \end{aligned}$$

for a certain \(P \in C^1[0,\infty )\). Consequently,

$$\begin{aligned} p(\varrho , \vartheta ) = \vartheta ^{\frac{5}{2}} P \left( \frac{\varrho }{\vartheta ^{\frac{3}{2}} } \right) + \frac{a}{3} \vartheta ^4,\ e(\varrho , \vartheta ) = \frac{3}{2} \frac{\vartheta ^{\frac{5}{2}} }{\varrho } P \left( \frac{\varrho }{\vartheta ^{\frac{3}{2}} } \right) + \frac{a}{\varrho } \vartheta ^4, \ a > 0.\nonumber \\ \end{aligned}$$
(2.3)

In addition, we suppose

$$\begin{aligned} P(0) = 0,\ P'(Z)> 0 \ \text{ for }\ Z \ge 0,\ 0 < \frac{ \frac{5}{3} P(Z) - P'(Z) Z }{Z} \le c \ \text{ for }\ Z > 0,\nonumber \\ \end{aligned}$$
(2.4)

that may be seen as a direct consequence of hypothesis of thermodynamic stability, see [11, Chapter 1], and Bechtel et al. [1]. It follows that the function \(Z \mapsto P(Z)/ Z^{\frac{5}{3}}\) is decreasing, and we suppose

$$\begin{aligned} \lim _{Z \rightarrow \infty } \frac{ P(Z) }{Z^{\frac{5}{3}}} = p_\infty > 0. \end{aligned}$$
(2.5)

The associated entropy takes the form

$$\begin{aligned} s(\varrho , \vartheta ) = \mathcal {S} \left( \frac{\varrho }{\vartheta ^{\frac{3}{2}} } \right) + \frac{4a}{3} \frac{\vartheta ^3}{\varrho }, \end{aligned}$$
(2.6)

where

$$\begin{aligned} \mathcal {S}'(Z) = -\frac{3}{2} \frac{ \frac{5}{3} P(Z) - P'(Z) Z }{Z^2} < 0. \end{aligned}$$
(2.7)

Finally, we impose the Third law of thermodynamics, see e.g. Belgiorno [2, 3], requiring the total entropy to vanish as soon as the absolute temperature approaches zero,

$$\begin{aligned} \lim _{Z \rightarrow \infty } \mathcal {S}(Z) = 0. \end{aligned}$$
(2.8)

It is easy to check that (2.4)–(2.8) imply

$$\begin{aligned} 0 \le \varrho \mathcal {S} \left( \frac{\varrho }{\vartheta ^{\frac{3}{2}}} \right) \le c \left( 1 + \varrho \log ^+(\varrho ) + \varrho \log ^+(\vartheta ) \right) . \end{aligned}$$
(2.9)

As for the transport coefficients, we suppose that they are continuously differentiable functions of the absolute temperature satisfying

$$\begin{aligned} 0< {\underline{\mu }}(1 + \vartheta )&\le \mu (\vartheta ),\ |\mu '(\vartheta )| \le \overline{\mu }, \nonumber \\ 0 \le \eta (\vartheta )&\le \overline{\eta }(1 + \vartheta ), \nonumber \\ 0 < {\underline{\kappa }} (1 + \vartheta ^\beta )&\le \kappa (\vartheta ) \le \overline{\kappa }(1 + \vartheta ^\beta ), \end{aligned}$$
(2.10)

where, in accordance with the existence theory developed in [7], we require

$$\begin{aligned} \beta > 6. \end{aligned}$$
(2.11)

2.2 Weak solutions

It is convenient to identify the time periodic functions (distributions) with objects defined on a periodic “flat sphere”

$$\begin{aligned} S_T = [0,T]|_{\{ 0, T \} }. \end{aligned}$$

We are ready to introduce the concept of time periodic solution to the Navier–Stokes–Fourier system (1.1)–(1.7).

figure d

The weak time–periodic solutions are therefore the weak solutions in the sense of [7] that are T-periodic in the time variable. The instantaneous values of the conservative variables \(\varrho (\tau , \cdot )\), \((\varrho \textbf{u})(\tau , \cdot )\) are well defined as well as the right and left-hand limits of the total entropy \(S = \varrho s(\varrho , \vartheta )\),

2.3 Main result

Having collected the necessary preliminary material we are ready to state our main result.

figure e

Remark 2.3

In the hypotheses of Theorem 2.2, we assume that \(\vartheta _B|_{\partial \Omega }\) is a restriction of a (smooth) function defined on the whole space \(R^d\).

The rest of the paper is devoted to the proof of Theorem 2.2.

3 Approximate problem

The most efficient way of constructing suitable approximate solutions seems adapting the result of [10] to the present setting. Specifically, the approximation scheme is based on penalization of the Dirichlet boundary condition for the temperature via the Robin boundary conditions

$$\begin{aligned} \textbf{q} \cdot \textbf{n} = \frac{1}{\varepsilon } |\vartheta - \vartheta _B|^{k}(\vartheta - \vartheta _B),\ k \ge 0, \ \text{ on }\ \partial \Omega , \end{aligned}$$
(3.1)

where \(\varepsilon > 0\) is a small parameter.

The approximate solutions \((\varrho _\varepsilon , \vartheta _\varepsilon , \textbf{u}_\varepsilon )\) are defined similarly to Definition 2.1:

  • Regularity class:

    $$\begin{aligned} \varrho _\varepsilon&\in C_{\textrm{weak}}(S_T; L^\gamma (\Omega )) \ \text{ for }\ \gamma = \frac{5}{3}, \nonumber \\ \textbf{u}_\varepsilon&\in L^2(S_T; W^{1,2}_0 (\Omega ; R^d)),\ \varrho _\varepsilon \textbf{u}_\varepsilon \in C_{\textrm{weak}}(S_T, L^{\frac{2 \gamma }{\gamma + 1}}(\Omega ; R^d)) \nonumber \\ \vartheta _\varepsilon ^{\beta /2} ,\ \log (\vartheta _\varepsilon )&\in L^2(S_T; W^{1,2}(\Omega )). \end{aligned}$$
    (3.2)

  • Equation of continuity:

    $$\begin{aligned} \int _{S_T} \int _{\Omega } \left[ \varrho _\varepsilon \partial _t \varphi + \varrho _\varepsilon \textbf{u}_\varepsilon \cdot \nabla _x\varphi \right] \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t&= 0, \end{aligned}$$
    (3.3)
    $$\begin{aligned} \int _{S_T} \int _{\Omega } \left[ b(\varrho _\varepsilon ) \partial _t \varphi + b(\varrho _\varepsilon ) \textbf{u}_\varepsilon \cdot \nabla _x\varphi + \Big ( b(\varrho _\varepsilon ) - b'(\varrho _\varepsilon ) \varrho _\varepsilon \Big ) {\textrm{div}}_x\textbf{u}_\varepsilon \varphi \right] \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t&=0 \end{aligned}$$
    (3.4)

    for any \(\varphi \in C^1(S_T \times \overline{\Omega } )\), and any \(b \in C^1(R)\), \(b' \in C_c(R)\).

  • Momentum equation:

    $$\begin{aligned}&\int _{S_T} \int _{\Omega } \Big [ \varrho _\varepsilon \textbf{u}_\varepsilon \cdot \partial _t \varvec{\varphi }+ \varrho _\varepsilon \textbf{u}_\varepsilon \otimes \textbf{u}_\varepsilon : \nabla _x\varvec{\varphi }- \varrho _\varepsilon (\varvec{\omega }\times \textbf{u}_\varepsilon ) \cdot \varvec{\varphi }+ p {\textrm{div}}_x\varvec{\varphi }\Big ] \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \nonumber \\ {}&= \int _{S_T} \int _{\Omega } \Big [ {\mathbb {S}} : \nabla _x\varvec{\varphi }- \varrho _\varepsilon \nabla _xG \cdot \varvec{\varphi }\Big ] \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \end{aligned}$$
    (3.5)

    for any \(\varvec{\varphi }\in C^1_c(S_T \times \Omega ; R^d)\).

  • Entropy inequality:

    $$\begin{aligned}&- \int _{S_T} \int _{\Omega } \left[ \varrho _\varepsilon s \partial _t \varphi + \varrho _\varepsilon s \textbf{u}_\varepsilon \cdot \nabla _x\varphi {+} \frac{\textbf{q}}{\vartheta _\varepsilon } \cdot \nabla _x\varphi \right] \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t {\ge } \int _{S_T} \int _{\Omega } \frac{\varphi }{\vartheta _\varepsilon } \left[ {\mathbb {S}} : {\mathbb {D}}_x\textbf{u}_\varepsilon {-} \frac{\textbf{q} \cdot \nabla _x\vartheta _\varepsilon }{\vartheta _\varepsilon } \right] \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \nonumber \\&\qquad + \frac{1}{\varepsilon } \int _{S_T} \int _{\partial \Omega } \varphi \frac{|\vartheta _B- \vartheta _\varepsilon |^k (\vartheta _B- \vartheta _\varepsilon ) }{\vartheta _\varepsilon } {\textrm{d}}\sigma _x \ \,{{\textrm{d}}} t \end{aligned}$$
    (3.6)

    for any \(\varphi \in C^1(S_T \times \overline{\Omega })\), \(\varphi \ge 0\).

  • Energy balance:

    $$\begin{aligned}&- \int _{S_T} \partial _t \psi \int _{\Omega } \left[ \frac{1}{2} \varrho _\varepsilon |\textbf{u}_\varepsilon |^2 + \varrho _\varepsilon e \right] \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t + \frac{1}{\varepsilon } \int _{S_T} \psi \int _{\partial \Omega } |\vartheta _\varepsilon - \vartheta _B|^k (\vartheta _\varepsilon - \vartheta _B) {\textrm{d}}\sigma _x \ \,{{\textrm{d}}} t \nonumber \\&\quad = \int _{S_T} \psi \int _{\Omega } \varrho _\varepsilon \textbf{u}_\varepsilon \cdot \nabla _xG \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \end{aligned}$$
    (3.7)

    for any \(\psi \in C^1(S_T)\),

cf. [10, Section 2.2].

3.1 Existence of approximate solutions

Our aim is to use the existence result proved in [10, Theorem 1] to obtain the approximate solutions \((\varrho _\varepsilon , \vartheta _\varepsilon , \textbf{u}_\varepsilon )_{\varepsilon > 0}\). To perform this step some comments are in order. In comparison with [10], the present problem features the following new ingredients:

  • The action of the Coriolis force in the momentum equation (3.5).

  • The function \(\vartheta _B\) in (3.1) is time dependent whereas its counterpart \(\Theta _0\) in [10] depends only on x.

  • The exponent k in (3.1) equals zero in [10].

It is easy to check that the existence proof in [10] can be modified to accommodate the above changes as soon as suitable a priori bounds similar to those in [10, Section 2.4] are established. To see this, we start with the energy balance (3.7) with \(\psi \equiv 1\) yielding

$$\begin{aligned}&\frac{1}{\varepsilon } \int _{S_T} \int _{\partial \Omega } |\vartheta _\varepsilon - \vartheta _B|^k (\vartheta _\varepsilon - \vartheta _B) {\textrm{d}}\sigma _x \ \,{{\textrm{d}}} t = \int _{S_T} \int _{\Omega } \varrho _\varepsilon \textbf{u}_\varepsilon \cdot \nabla _xG \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t = - \int _{S_T} \int _{\Omega } \varrho _\varepsilon \partial _t G \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \nonumber \\&\quad \le M_0 \Vert \partial _t G \Vert _{L^\infty (S_T \times \Omega )} ,\ \text{ where }\ M_0 = \int _{\Omega } \varrho _\varepsilon \ \,{\textrm{d}} {x}. \end{aligned}$$
(3.8)

As \(\vartheta _\varepsilon > 0\) a.a., (3.8) yields the bound

$$\begin{aligned} \Vert \vartheta _\varepsilon \Vert _{L^{k+1}(S_T \times \partial \Omega )} {\mathop {\sim }\limits ^{<}}1 \end{aligned}$$
(3.9)

in terms of the data and uniform for \(\varepsilon \rightarrow 0\). Consequently, the entropy inequality (3.6) gives rise to the bound on the entropy production rate

$$\begin{aligned} \int _{S_T} \int _{\Omega } \frac{1}{\vartheta _\varepsilon } \left[ {\mathbb {S}} : {\mathbb {D}}_x\textbf{u}_\varepsilon - \frac{\textbf{q} \cdot \nabla _x\vartheta _\varepsilon }{\vartheta _\varepsilon } \right] \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \le { c(\varepsilon , G, \vartheta _B, M_0) } \end{aligned}$$
(3.10)

and the remaining estimates are obtained exactly as in [10, Section 2.4]. Note that the right–hand side of (3.10) may blow up for \(\varepsilon \rightarrow 0\).

With the necessary a priori bounds at hand, we obtain a family of approximate solutions \((\varrho _\varepsilon , \vartheta _\varepsilon , \textbf{u}_\varepsilon )_{\varepsilon > 0}\) exactly as in [10].

Proposition 3.1

(Approximate solutions) In addition to the hypotheses of Theorem 2.2, let

$$\begin{aligned} \varepsilon> 0,\ 6 < k + 1 = \beta , \ M_0 > 0 \end{aligned}$$
(3.11)

be given.

Then the approximate problem (3.2)–(3.7) admits a solution \((\varrho _\varepsilon , \vartheta _\varepsilon , \textbf{u}_\varepsilon )\).

3.2 Approximate ballistic energy balance

Let \({\tilde{\vartheta }}\in C^1(S_T \times \overline{\Omega })\) satisfy (2.18). Choosing \(\varphi (t,x) = \psi (t) {\tilde{\vartheta }}(t,x)\), where \(\psi \in C^1(S_T)\), \(\psi \ge 0\), as a test function in the approximate entropy inequality (3.6) and adding the resulting integral to the energy balance (3.7), we deduce

$$\begin{aligned}&- \int _{S_T} \partial _t \psi \int _{\Omega } \left[ \frac{1}{2} \varrho _\varepsilon |\textbf{u}_\varepsilon |^2 + \varrho _\varepsilon e - {\tilde{\vartheta }}\varrho _\varepsilon s \right] \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t + \int _{S_T} \psi \int _{\Omega } \frac{{\tilde{\vartheta }}}{\vartheta _\varepsilon } \left[ {\mathbb {S}}: {\mathbb {D}}_x\textbf{u}_\varepsilon - \frac{\textbf{q} \cdot \nabla _x\vartheta _\varepsilon }{\vartheta _\varepsilon } \right] \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \nonumber \\&\qquad + \frac{1}{\varepsilon } \int _{S_T} \psi \int _{\partial \Omega } \frac{ |\vartheta _\varepsilon - \vartheta _B|^{k + 2} }{\vartheta _\varepsilon } \ {\textrm{d}}\sigma _x \ \,{{\textrm{d}}} t \nonumber \\&\quad \le \int _{S_T} \psi \int _{\Omega } \left[ \varrho _\varepsilon \textbf{u}_\varepsilon \cdot \nabla _xG - \varrho _\varepsilon s \textbf{u}_\varepsilon \cdot \nabla _x{\tilde{\vartheta }}- \frac{\textbf{q}}{\vartheta _\varepsilon } \cdot \nabla _x{\tilde{\vartheta }}- \partial _t {\tilde{\vartheta }}\varrho _\varepsilon s \right] \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t . \end{aligned}$$
(3.12)

Inequality (3.12) is obviously a counter part of the ballistic energy balance (2.17) and will be used in the forthcoming part to deduce the necessary bounds on the family of approximate solutions.

4 Uniform bounds

In order to perform the limit \(\varepsilon \rightarrow 0\) in the family of approximate solutions obtained in Proposition 3.1, we need uniform bounds independent of \(\varepsilon \).

4.1 Mass conservation

Obviously, as the total mass of the fluid is conserved, we get

$$\begin{aligned} M_0 = \int _{\Omega } \varrho _\varepsilon (t,\cdot ) \ \,{\textrm{d}} {x} \ \text{ for } \text{ all }\ t \in S_T \ \Rightarrow \ \sup _{t \in S_T} \Vert \varrho _\varepsilon (t, \cdot )\Vert _{L^1(\Omega )} {\mathop {\sim }\limits ^{<}}1. \end{aligned}$$
(4.1)

4.2 Energy estimates

As both \(\partial \Omega \) and the boundary data \(\vartheta _B\) are smooth, we may suppose that

$$\begin{aligned} \Delta _x\vartheta _B(t, \cdot ) = 0 \ \text{ in }\ \Omega \ \text{ for } \text{ any }\ t \in S_T. \end{aligned}$$
(4.2)

Choosing \(\psi = 1\), \({\tilde{\vartheta }}= \vartheta _B\) in the ballistic energy inequality (3.12) we get

$$\begin{aligned}&\int _{S_T} \int _{\Omega } \frac{\vartheta _B}{\vartheta _\varepsilon } \left[ {\mathbb {S}}({\mathbb {D}}_x\textbf{u}_\varepsilon ) : {\mathbb {D}}_x\textbf{u}_\varepsilon + \frac{\kappa (\vartheta _\varepsilon ) |\nabla _x\vartheta _\varepsilon |^2 }{\vartheta _\varepsilon } \right] \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t +\frac{1}{\varepsilon } \int _{S_T} \int _{\partial \Omega } \frac{ |\vartheta _\varepsilon - \vartheta _B|^{k + 2} }{\vartheta _\varepsilon } \ {\textrm{d}}\sigma _x \ \,{{\textrm{d}}} t \nonumber \\&\quad \le \int _{S_T} \int _{\Omega } \left[ \varrho _\varepsilon \textbf{u}_\varepsilon \cdot \nabla _xG - \varrho _\varepsilon s(\varrho _\varepsilon , \vartheta _\varepsilon ) \textbf{u}_\varepsilon \cdot \nabla _x\vartheta _B+ \frac{\kappa (\vartheta _\varepsilon ) \nabla _x\vartheta _\varepsilon }{\vartheta _\varepsilon } \cdot \nabla _x\vartheta _B- \partial _t {\tilde{\vartheta }}\varrho _\varepsilon s(\varrho _\varepsilon , \vartheta _\varepsilon ) \right] \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t . \end{aligned}$$
(4.3)

By virtue of hypothesis (2.10) and Korn’s inequality, we obtain

$$\begin{aligned} \Vert \textbf{u}_\varepsilon \Vert _{W^{1,2}_0(\Omega ; R^d)}^2 {\mathop {\sim }\limits ^{<}}\int _{\Omega } \frac{\vartheta _B}{\vartheta _\varepsilon } {\mathbb {S}}({\mathbb {D}}_x\textbf{u}_\varepsilon ) : {\mathbb {D}}_x\textbf{u}_\varepsilon \ \,{\textrm{d}} {x}. \end{aligned}$$

Moreover, again by virtue of (2.10),

$$\begin{aligned} \int _{\Omega } \left[ | \nabla _x\vartheta _\varepsilon ^{\frac{\beta }{2}} |^2 + |\nabla _x\log (\vartheta _\varepsilon ) |^2 \right] \ \,{\textrm{d}} {x} {\mathop {\sim }\limits ^{<}}\int _{\Omega } \frac{\vartheta _B}{\vartheta _\varepsilon } \frac{\kappa (\vartheta _\varepsilon ) |\nabla _x\vartheta _\varepsilon |^2 }{\vartheta _\varepsilon } \ \,{\textrm{d}} {x}. \end{aligned}$$

By Poincarè inequality (see e.g. Theorem 4.4.6 in [27]) we obtain that

$$\begin{aligned}{} & {} \int _{\Omega } | \vartheta _\varepsilon ^{\frac{\beta }{2}} |^2 \ \,{\textrm{d}} {x} {\mathop {\sim }\limits ^{<}}\int _{\partial \Omega }|\vartheta _\varepsilon ^{\frac{\beta }{2}}|^2\ {\textrm{d}}\sigma _x+\int _{\Omega } \left| \vartheta _\varepsilon ^{\frac{\beta }{2}}- \int _{\partial \Omega } \vartheta _\varepsilon ^{\frac{\beta }{2}}\ {\textrm{d}}\sigma _x\right| ^2 \ \,{\textrm{d}} {x} {\mathop {\sim }\limits ^{<}}\int _{\partial \Omega }|\vartheta _\varepsilon ^{\frac{\beta }{2}}|^2\ {\textrm{d}}\sigma _x\\{} & {} \quad +\int _{\Omega } | \nabla _x\vartheta _\varepsilon ^{\frac{\beta }{2}} |^2 \ \,{\textrm{d}} {x}, \end{aligned}$$

as well as

$$\begin{aligned} \int _{\Omega } | \log (\vartheta _\varepsilon ) |^2 \ \,{\textrm{d}} {x} {\mathop {\sim }\limits ^{<}}\int _{\partial \Omega }|\log (\vartheta _\varepsilon )|^2\ {\textrm{d}}\sigma _x+\int _{\Omega } | \nabla _x\log (\vartheta _\varepsilon ) |^2 \ \,{\textrm{d}} {x}. \end{aligned}$$

Collecting the last three inequalities, hypothesis (3.11) and estimating the boundary terms

$$\begin{aligned} |\vartheta _\varepsilon ^{\frac{\beta }{2}}|^2+|\log (\vartheta _\varepsilon )|^2{\mathop {\sim }\limits ^{<}}\frac{ |\vartheta _\varepsilon - \vartheta _B|^{k + 2} }{\vartheta _\varepsilon }{\mathop {\sim }\limits ^{<}}\frac{1}{2\varepsilon }\frac{ |\vartheta _\varepsilon - \vartheta _B|^{k + 2} }{\vartheta _\varepsilon } \end{aligned}$$

gives

$$\begin{aligned}{} & {} \left\| \vartheta _\varepsilon ^{\frac{\beta }{2}} \right\| _{W^{1,2}(\Omega )}^2 + \left\| \log (\vartheta _\varepsilon ) \right\| _{W^{1,2}(\Omega )}^2\\{} & {} \quad {\mathop {\sim }\limits ^{<}}\left( 1 + \frac{1}{2\varepsilon } \int _{\partial \Omega } \frac{ |\vartheta _\varepsilon - \vartheta _B|^{k + 2} }{\vartheta _\varepsilon } \ {\textrm{d}}\sigma _x + \int _{\Omega } \frac{\vartheta _B}{\vartheta _\varepsilon } \frac{\kappa (\vartheta _\varepsilon ) |\nabla _x\vartheta _\varepsilon |^2 }{\vartheta _\varepsilon } \ \,{\textrm{d}} {x} \right) . \end{aligned}$$

Gathering the previous observations, we may infer that

$$\begin{aligned}&\int _{S_T} \left[ \Vert \textbf{u}_\varepsilon \Vert _{W^{1,2}_0(\Omega ; R^d)}^2 + \left\| \vartheta _\varepsilon ^{\frac{\beta }{2}} \right\| _{W^{1,2}(\Omega )}^2 + \left\| \log (\vartheta _\varepsilon ) \right\| _{W^{1,2}(\Omega )}^2 \right] \ \,{{\textrm{d}}} t +\frac{1}{\varepsilon } \int _{S_T} \int _{\partial \Omega } \frac{ |\vartheta _\varepsilon - \vartheta _B|^{\beta + 1} }{\vartheta _\varepsilon } \ {\textrm{d}}\sigma _x \ \,{{\textrm{d}}} t \nonumber \\&\quad {\mathop {\sim }\limits ^{<}}\left( 1 + \left| \int _{S_T} \int _{\Omega } \left[ \varrho _\varepsilon \textbf{u}_\varepsilon \cdot \nabla _xG - \varrho _\varepsilon s(\varrho _\varepsilon , \vartheta _\varepsilon ) \textbf{u}_\varepsilon \cdot \nabla _x\vartheta _B+ \frac{\kappa (\vartheta _\varepsilon ) \nabla _x\vartheta _\varepsilon }{\vartheta _\varepsilon } \cdot \nabla _x\vartheta _B- \partial _t {\tilde{\vartheta }}\varrho _\varepsilon s(\varrho _\varepsilon , \vartheta _\varepsilon )\right] \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \right| \right) . \end{aligned}$$
(4.4)

Now, as \(\varrho _\varepsilon \), \(\textbf{u}_\varepsilon \) solve the equation of continuity (2.13),

$$\begin{aligned} \int _{S_T} \int _{\Omega } \varrho _\varepsilon \textbf{u}_\varepsilon \cdot \nabla _xG \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t = - \int _{S_T} \varrho _\varepsilon \partial _t G \ \,{{\textrm{d}}} t \le c(M_0, G). \end{aligned}$$

In addition, denoting

$$\begin{aligned} \mathcal {K}(\vartheta ) = \int _1^\vartheta \frac{\kappa (z) }{z} \ {\textrm{d}}z, \end{aligned}$$

we obtain, by virtue of (4.2),

$$\begin{aligned} \int _{\Omega } \frac{\kappa (\vartheta _\varepsilon ) \nabla _x\vartheta _\varepsilon }{\vartheta _\varepsilon } \cdot \nabla _x\vartheta _B \ \,{\textrm{d}} {x} = \int _{\Omega } \nabla _x\mathcal {K} (\vartheta _\varepsilon ) \cdot \nabla _x\vartheta _B \ \,{\textrm{d}} {x} = \int _{\partial \Omega } \mathcal {K} (\vartheta _\varepsilon ) \nabla _x\vartheta _B\cdot \textbf{n} {\textrm{d}}\sigma _x. \end{aligned}$$

Consequently, as \(\kappa \) satisfies hypothesis (2.10), we conclude

$$\begin{aligned} \left| \int _{\Omega } \frac{\kappa (\vartheta _\varepsilon ) \nabla _x\vartheta _\varepsilon }{\vartheta _\varepsilon } \cdot \nabla _x\vartheta _B \ \,{\textrm{d}} {x} \right|= & {} \left| \int _{\partial \Omega } \mathcal {K} (\vartheta _\varepsilon ) \nabla _x\vartheta _B\cdot \textbf{n} {\textrm{d}}\sigma _x \right| \\ {}&{\mathop {\sim }\limits ^{<}}&\left( 1+ { \int _{\partial \Omega } \frac{ |\vartheta _\varepsilon - \vartheta _B|^{\beta + 1} }{\vartheta _\varepsilon } \ {\textrm{d}}\sigma _x}\right) . \end{aligned}$$

Thus inequality (4.4) reduces to

$$\begin{aligned}&\int _{S_T} \left[ \Vert \textbf{u}_\varepsilon \Vert _{W^{1,2}_0(\Omega ; R^d)}^2 + \left\| \vartheta _\varepsilon ^{\frac{\beta }{2}} \right\| _{W^{1,2}(\Omega )}^2 + \left\| \log (\vartheta _\varepsilon ) \right\| _{W^{1,2}(\Omega )}^2 \right] \ \,{{\textrm{d}}} t \nonumber \\&\qquad +\frac{1}{\varepsilon } \int _{S_T} \int _{\partial \Omega } \frac{ |\vartheta _\varepsilon - \vartheta _B|^{\beta + 1} }{\vartheta _\varepsilon } \ {\textrm{d}}\sigma _x \ \,{{\textrm{d}}} t \nonumber \\&\quad {\mathop {\sim }\limits ^{<}}\left( 1 + \int _{S_T} \int _{\Omega } \left( | \varrho _\varepsilon s(\varrho _\varepsilon , \vartheta _\varepsilon ) \textbf{u}_\varepsilon \cdot \nabla _x\vartheta _B| + |\partial _t {\tilde{\vartheta }}\varrho _\varepsilon s(\varrho _\varepsilon , \vartheta _\varepsilon )|\right) \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \right) . \end{aligned}$$
(4.5)

In accordance with hypothesis (2.6), we decompose the entropy as

$$\begin{aligned} \varrho _\varepsilon s(\varrho _\varepsilon , \vartheta _\varepsilon ) = \varrho _\varepsilon \mathcal {S} \left( \frac{\varrho _\varepsilon }{\vartheta _\varepsilon ^{\frac{3}{2}} } \right) + \frac{4a}{3} \vartheta _\varepsilon ^3. \end{aligned}$$

Consequently, the radiation component may be handled as

$$\begin{aligned} \int _{\Omega } | \vartheta _\varepsilon ^3 \textbf{u}_\varepsilon \cdot \nabla _x\vartheta _B| \ \,{\textrm{d}} {x} \le \delta \Vert \textbf{u}_\varepsilon \Vert _{L^2(\Omega ; R^d)}^2 + c(\delta , \vartheta _B) \int _{\Omega } \vartheta _\varepsilon ^6 \ \,{\textrm{d}} {x} \end{aligned}$$

for any \(\delta > 0\). Consequently, as \(\beta > 6\), this term can be absorbed by the left–hand side of (4.5) yielding

$$\begin{aligned}&\int _{S_T} \left[ \Vert \textbf{u}_\varepsilon \Vert _{W^{1,2}_0(\Omega ; R^d)}^2 + \left\| \vartheta _\varepsilon ^{\frac{\beta }{2}} \right\| _{W^{1,2}(\Omega )}^2 + \left\| \log (\vartheta _\varepsilon ) \right\| _{W^{1,2}(\Omega )}^2 \right] \ \,{{\textrm{d}}} t \nonumber \\&\qquad +\frac{1}{\varepsilon } \int _{S_T} \int _{\partial \Omega } \frac{ |\vartheta _\varepsilon - \vartheta _B|^{\beta + 1} }{\vartheta _\varepsilon } \ {\textrm{d}}\sigma _x \ \,{{\textrm{d}}} t \nonumber \\ {}&\quad {\mathop {\sim }\limits ^{<}}\left( 1 + \int _{S_T} \int _{\Omega } \left| \varrho _\varepsilon \mathcal {S} \left( \frac{\varrho _\varepsilon }{\vartheta _\varepsilon ^{\frac{3}{2}} } \right) \textbf{u}_\varepsilon \cdot \nabla _x\vartheta _B\right| +\left| \varrho _\varepsilon \mathcal {S} \left( \frac{\varrho _\varepsilon }{\vartheta _\varepsilon ^{\frac{3}{2}} } \right) \partial _t {\tilde{\vartheta }}\right| \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \right) . \end{aligned}$$
(4.6)

Finally, following the arguments of [14, Section 4.4], we make use of the Third law of thermodynamics enforced through hypothesis (2.8). Specifically, if

$$\begin{aligned} \frac{\varrho }{\vartheta ^{\frac{3}{2}}}< r \ \text{ meaning }\ \varrho < r \vartheta ^{\frac{3}{2}}, \end{aligned}$$

we get, by virtue of (2.9),

$$\begin{aligned} 0 \le \varrho \mathcal {S} \left( \frac{\varrho }{\vartheta ^{\frac{3}{2}}} \right) {\mathop {\sim }\limits ^{<}}\left( 1 + r \vartheta ^{\frac{3}{2}} \left[ \log ^+ (r \vartheta ^{\frac{3}{2}} ) + \log ^+(\vartheta ) \right] \right) . \end{aligned}$$
(4.7)

Consequently, we deduce from (4.6),

(4.8)

where \(\Lambda (r) \rightarrow \infty \) as \(r \rightarrow \infty \).

Now, again by hypothesis (2.8),

$$\begin{aligned} 0 \le \mathbb {1}_{ \left\{ \frac{\varrho _\varepsilon }{\vartheta _\varepsilon ^{\frac{3}{2}}} \ge r \right\} }\mathcal {S} \left( \frac{\varrho _\varepsilon }{\vartheta _\varepsilon ^{\frac{3}{2}} } \right) \le \mathcal {S}(r) \rightarrow 0 \ \text{ as }\ r \rightarrow \infty . \end{aligned}$$

In an analogous way we treat the term \(\int _{S_T} \int _{\Omega } \partial _t {\tilde{\vartheta }}\varrho _\varepsilon s \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \). Going back to (4.8) we conclude

$$\begin{aligned}&\int _{S_T} \left[ \Vert \textbf{u}_\varepsilon \Vert _{W^{1,2}_0(\Omega ; R^d)}^2 + \left\| \vartheta _\varepsilon ^{\frac{\beta }{2}} \right\| _{W^{1,2}(\Omega )}^2 + \left\| \log (\vartheta _\varepsilon ) \right\| _{W^{1,2}(\Omega )}^2 \right] \ \,{{\textrm{d}}} t \nonumber \\&\quad +\frac{1}{\varepsilon } \int _{S_T} \int _{\partial \Omega } \frac{ |\vartheta _\varepsilon - \vartheta _B|^{\beta + 1} }{\vartheta _\varepsilon } \ {\textrm{d}}\sigma _x \ \,{{\textrm{d}}} t \nonumber \\&\quad {\mathop {\sim }\limits ^{<}}\left( \Lambda (r) + \mathcal {S}(r) \int _{S_T} \int _{\Omega } \left( \left| \varrho _\varepsilon \textbf{u}_\varepsilon \right| +\varrho _\varepsilon \right) \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \right) , \nonumber \\&\quad \Lambda (r) \rightarrow \infty ,\ \mathcal {S}(r) \rightarrow 0 \ \text{ as }\ r \rightarrow \infty . \end{aligned}$$
(4.9)

4.3 Pressure estimates

To close the estimates we have to control the density in terms of the integrals on the right–hand side of (4.9). To this end, we use the nowadays standard pressure estimates obtained via Bogovskii operator. Specifically, we use the quantity

$$\begin{aligned} \varphi (t,x) = \mathcal {B} \left[ \varrho _\varepsilon ^\omega - \frac{1}{|\Omega |} \int _{\Omega } \varrho _\varepsilon ^\omega \ \,{\textrm{d}} {x} \right] , \omega > 0, \end{aligned}$$

as a test function in the momentum equation (3.5). Here \(\mathcal {B}\) denotes the operator enjoying the following properties:

  • $$\begin{aligned} \mathcal {B} : L^q_0 (\Omega ) \equiv \left\{ v \in L^q(\Omega ) \ \Big |\ \int _{\Omega } v \ \,{\textrm{d}} {x} = 0 \right\} \rightarrow W^{1,q}_0 (\Omega ; R^d) ,\ 1< q < \infty ;\nonumber \\ \end{aligned}$$
    (4.10)
  • $$\begin{aligned} {\textrm{div}}_x\mathcal {B} [v] = v; \end{aligned}$$

  • if \(v = {\textrm{div}}_x\textbf{g}\), with \(\textbf{g} \in L^q(\Omega ; R^d)\), \({\textrm{div}}_x\textbf{g} \in L^r(\Omega )\), \(\textbf{g} \cdot \textbf{n}|_{\partial \Omega } = 0\), then

    $$\begin{aligned} \Vert \mathcal {B}[ {\textrm{div}}_x\textbf{g} ]\Vert _{L^r(\Omega ; R^d)} {\mathop {\sim }\limits ^{<}}\Vert \textbf{g} \Vert _{L^r(\Omega ; R^d)}, \end{aligned}$$
    (4.11)

see Galdi [15, Chapter 3] or Geißert et al. [16]. Boundedness of the operator \(\mathcal {B}\) stated in (4.10), (4.11) will be systematically used in the estimates below.

After a straightforward manipulation (see e.g. [9]), we obtain

$$\begin{aligned}&\int _{S_T} \int _{\Omega } p(\varrho _\varepsilon , \vartheta _\varepsilon ) \varrho _\varepsilon ^\omega \ \,{\textrm{d}} {x} \,{{\textrm{d}}} t \nonumber \\ {}&\quad = \int _{S_T} \frac{1}{|\Omega |} \left( \int _{\Omega } \varrho _\varepsilon ^\omega \ \,{\textrm{d}} {x} \right) \left( \int _{\Omega } p(\varrho _\varepsilon , \vartheta _\varepsilon ) \ \,{\textrm{d}} {x} \right) \ \,{{\textrm{d}}} t \nonumber \\&\qquad -\int _{S_T} \int _{\Omega } \varrho _\varepsilon (\textbf{u}_\varepsilon \otimes \textbf{u}_\varepsilon ): \nabla _x\mathcal {B} \left[ \varrho _\varepsilon ^\omega - \frac{1}{|\Omega |} \int _{\Omega } \varrho _\varepsilon ^\omega \ \,{\textrm{d}} {x} \right] \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \nonumber \\&\qquad + \int _{S_T} \int _{\Omega } \varrho _\varepsilon (\varvec{\omega }\times \textbf{u}_\varepsilon ) \cdot \mathcal {B} \left[ \varrho _\varepsilon ^\omega - \frac{1}{|\Omega |} \int _{\Omega } \varrho _\varepsilon ^\omega \ \,{\textrm{d}} {x} \right] \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \nonumber \\&\qquad + \int _{S_T} \int _{\Omega } {\mathbb {S}}(\vartheta _\varepsilon , {\mathbb {D}}_x\textbf{u}_\varepsilon ) : \nabla _x\mathcal {B} \left[ \varrho _\varepsilon ^\omega - \frac{1}{|\Omega |} \int _{\Omega } \varrho _\varepsilon ^\omega \ \,{\textrm{d}} {x} \right] \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \nonumber \\&\qquad - \int _{S_T} \int _{\Omega } \varrho _\varepsilon \nabla _xG \cdot \mathcal {B} \left[ \varrho _\varepsilon ^\omega - \frac{1}{|\Omega |} \int _{\Omega } \varrho _\varepsilon ^\omega \ \,{\textrm{d}} {x} \right] \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \nonumber \\&\qquad + \int _{S_T} \int _{\Omega } \varrho _\varepsilon \textbf{u}_\varepsilon \cdot \mathcal {B}[{\textrm{div}}_x( \varrho _\varepsilon ^\omega \textbf{u}_\varepsilon ) ] \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \nonumber \\&\qquad + (\omega - 1) \int _{S_T} \int _{\Omega } \varrho _\varepsilon \textbf{u}_\varepsilon \cdot \mathcal {B} \left[ \varrho _\varepsilon ^\omega {\textrm{div}}_x\textbf{u}_\varepsilon - \frac{1}{|\Omega |} \int _{\Omega } \varrho _\varepsilon ^\omega {\textrm{div}}_x\textbf{u}_\varepsilon \ \,{\textrm{d}} {x} \right] \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t . \end{aligned}$$
(4.12)

Since the total mass \(M_0\) is constant, the smoothing properties of \(\mathcal {B}\) yield

$$\begin{aligned} \left\| \mathcal {B} \left[ \varrho _\varepsilon ^\omega - \frac{1}{|\Omega |} \int _{\Omega } \varrho _\varepsilon ^\omega \ \,{\textrm{d}} {x} \right] \right\| _{L^\infty (S_T \times \Omega ; R^d) } \le c(M_0) \ \text{ as } \text{ soon } \text{ as }\ \omega < \frac{1}{d}. \end{aligned}$$

Moreover, in accordance with hypotheses (2.3)–(2.5),

$$\begin{aligned} \varrho ^{\frac{5}{3}} + \vartheta ^4 {\mathop {\sim }\limits ^{<}}p(\varrho , \vartheta ) {\mathop {\sim }\limits ^{<}}\varrho ^{\frac{5}{3}} + \vartheta ^4 + 1. \end{aligned}$$

In view of these facts, inequality (4.12) gives rise to

$$\begin{aligned} \int _{S_T}&\int _{\Omega } \varrho _\varepsilon ^{\frac{5}{3} + \omega } \ \,{\textrm{d}} {x} \,{{\textrm{d}}} t \le c(M_0) \Big ( 1 + \int _{S_T} \int _{\Omega } \vartheta _\varepsilon ^4 \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \nonumber \\&-\int _{S_T} \int _{\Omega } \varrho _\varepsilon (\textbf{u}_\varepsilon \otimes \textbf{u}_\varepsilon ): \nabla _x\mathcal {B} \left[ \varrho _\varepsilon ^\omega - \frac{1}{|\Omega |} \int _{\Omega } \varrho _\varepsilon ^\omega \ \,{\textrm{d}} {x} \right] \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \nonumber \\&+ \int _{S_T} \int _{\Omega } \varrho _\varepsilon (\varvec{\omega }\times \textbf{u}_\varepsilon ) \cdot \mathcal {B} \left[ \varrho _\varepsilon ^\omega - \frac{1}{|\Omega |} \int _{\Omega } \varrho _\varepsilon ^\omega \ \,{\textrm{d}} {x} \right] \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \nonumber \\&+ \int _{S_T} \int _{\Omega } {\mathbb {S}}(\vartheta _\varepsilon , {\mathbb {D}}_x\textbf{u}_\varepsilon ) : \nabla _x\mathcal {B} \left[ \varrho _\varepsilon ^\omega - \frac{1}{|\Omega |} \int _{\Omega } \varrho _\varepsilon ^\omega \ \,{\textrm{d}} {x} \right] \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \nonumber \\&+ \int _{S_T} \int _{\Omega } \varrho _\varepsilon \textbf{u}_\varepsilon \cdot \mathcal {B}[{\textrm{div}}_x( \varrho _\varepsilon ^\omega \textbf{u}_\varepsilon ) ] \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \nonumber \\&+ (\omega - 1) \int _{S_T} \int _{\Omega } \varrho _\varepsilon \textbf{u}_\varepsilon \cdot \mathcal {B} \left[ \varrho _\varepsilon ^\omega {\textrm{div}}_x\textbf{u}_\varepsilon - \frac{1}{|\Omega |} \int _{\Omega } \varrho _\varepsilon ^\omega {\textrm{div}}_x\textbf{u}_\varepsilon \ \,{\textrm{d}} {x} \right] \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \Big ). \end{aligned}$$
(4.13)

The following steps will be performed for \(d=3\). Obviously even better estimates can be obtained if \(d=2\). First,

$$\begin{aligned}&\left| \int _{S_T} \int _{\Omega } \varrho _\varepsilon (\textbf{u}_\varepsilon \otimes \textbf{u}_\varepsilon ): \nabla _x\mathcal {B} \left[ \varrho _\varepsilon ^\omega - \frac{1}{|\Omega |} \int _{\Omega } \varrho _\varepsilon ^\omega \ \,{\textrm{d}} {x} \right] \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \right| \\&\quad {\mathop {\sim }\limits ^{<}}\int _{S_T} \Vert \varrho _\varepsilon \Vert _{L^\gamma (\Omega )} \Vert \textbf{u}_\varepsilon \Vert ^2_{L^6(\Omega ; R^3)} \Vert \varrho _\varepsilon ^\omega \Vert _{L^q(\Omega )} \ \,{{\textrm{d}}} t \\ {}&\quad {\mathop {\sim }\limits ^{<}}\sup _{t \in S_T} \Vert \varrho _\varepsilon \Vert _{L^\gamma (\Omega )} \int _{S_T} \Vert \textbf{u}_\varepsilon \Vert ^2_{W^{1,2}_0(\Omega ; R^3)} \ \,{{\textrm{d}}} t \sup _{t \in S_T} \Vert \varrho ^\omega \Vert _{L^q (\Omega )} \,{{\textrm{d}}} t , \end{aligned}$$

where

$$\begin{aligned} q = \frac{3 \gamma }{2 \gamma - 3}> 1 \ \text{ provided }\ \gamma > \frac{3}{2}. \end{aligned}$$

Fixing

$$\begin{aligned} \gamma = \frac{5}{3},\ \omega = \frac{3 \gamma }{2 \gamma - 3} = \frac{1}{15} \end{aligned}$$
(4.14)

and using the fact that the total mass \(M_0\) is conserved, we get

$$\begin{aligned}&\left| \int _{S_T} \int _{\Omega } \varrho _\varepsilon (\textbf{u}_\varepsilon \otimes \textbf{u}_\varepsilon ): \nabla _x\mathcal {B} \left[ \varrho _\varepsilon ^\omega - \frac{1}{|\Omega |} \int _{\Omega } \varrho _\varepsilon ^\omega \ \,{\textrm{d}} {x} \right] \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \right| \\ {}&\quad \le c(M_0) \sup _{t \in S_T} \Vert \varrho _\varepsilon \Vert _{L^\gamma (\Omega )} \int _{S_T} \Vert \textbf{u}_\varepsilon \Vert ^2_{W^{1,2}_0(\Omega ; R^3)} \ \,{{\textrm{d}}} t . \end{aligned}$$

Seeing that the integral containing the Coriolis force can be controlled in a similar way we may rewrite (4.13) in the form

$$\begin{aligned} \int _{S_T}&\int _{\Omega } \varrho _\varepsilon ^{\frac{5}{3} + \omega } \ \,{\textrm{d}} {x} \,{{\textrm{d}}} t \le c(M_0) \Big ( 1 + \int _{S_T} \int _{\Omega } \vartheta _\varepsilon ^4 \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \nonumber \\&+ \sup _{t \in S_T} \Vert \varrho _\varepsilon \Vert _{L^\gamma (\Omega )} \int _{S_T} \Vert \textbf{u}_\varepsilon \Vert ^2_{W^{1,2}_0(\Omega ; R^3)} \ \,{{\textrm{d}}} t \nonumber \\&+ \int _{S_T} \int _{\Omega } {\mathbb {S}}(\vartheta _\varepsilon , {\mathbb {D}}_x\textbf{u}_\varepsilon ) : \nabla _x\mathcal {B} \left[ \varrho _\varepsilon ^\omega - \frac{1}{|\Omega |} \int _{\Omega } \varrho _\varepsilon ^\omega \ \,{\textrm{d}} {x} \right] \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \nonumber \\&+ \int _{S_T} \int _{\Omega } \varrho _\varepsilon \textbf{u}_\varepsilon \cdot \mathcal {B}[{\textrm{div}}_x( \varrho _\varepsilon ^\omega \textbf{u}_\varepsilon ) ] \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \nonumber \\&+ (\omega - 1) \int _{S_T} \int _{\Omega } \varrho _\varepsilon \textbf{u}_\varepsilon \cdot \mathcal {B} \left[ \varrho _\varepsilon ^\omega {\textrm{div}}_x\textbf{u}_\varepsilon - \frac{1}{|\Omega |} \int _{\Omega } \varrho _\varepsilon ^\omega {\textrm{div}}_x\textbf{u}_\varepsilon \ \,{\textrm{d}} {x} \right] \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \Big ). \end{aligned}$$
(4.15)

In a similar way, we get

$$\begin{aligned}&\left| \int _{S_T} \int _{\Omega } \varrho _\varepsilon \textbf{u}_\varepsilon \cdot \mathcal {B}[{\textrm{div}}_x(\varrho _\varepsilon ^\omega \textbf{u}_\varepsilon )] \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \right| \\&\quad {\mathop {\sim }\limits ^{<}}\int _{S_T} \Vert \varrho _\varepsilon \Vert _{L^\gamma (\Omega )} \Vert \textbf{u}_\varepsilon \Vert _{L^6(\Omega ; R^3)} \Vert \varrho _\varepsilon ^\omega \textbf{u}_\varepsilon \Vert _{L^q(\Omega ; R^3)} \ \,{{\textrm{d}}} t , \end{aligned}$$

where

$$\begin{aligned} \frac{1}{\gamma }+ \frac{1}{6} + \frac{1}{q} = 1. \end{aligned}$$

In addition,

$$\begin{aligned} \Vert \varrho _\varepsilon ^\omega \textbf{u}_\varepsilon \Vert _{L^q(\Omega ; R^3)} \le \Vert \textbf{u}_\varepsilon \Vert _{L^6(\Omega ; R^3)} \Vert \varrho _\varepsilon ^\omega \Vert _{L^p(\Omega )}, \text{ where }\ \frac{1}{q} = \frac{1}{6} + \frac{1}{p}; \end{aligned}$$

whence

$$\begin{aligned} \left| \int _{\tau }^{\tau + 1} \int _{\Omega } \varrho \textbf{u}\cdot \mathcal {B}[{\textrm{div}}_x(\varrho ^\omega \textbf{u})] \ \,{\textrm{d}} {x} \,dt \right| \le c(M) \sup _{t \in (\tau , \tau + 1)} \Vert \varrho \Vert _{L^{\frac{5}{3}} (\Omega )} \int _{\tau }^{\tau + 1} \Vert \textbf{u}\Vert ^2_{W^{1,2}_0(\Omega ; R^3)} \,{{\textrm{d}}} t \end{aligned}$$

as soon as (4.14) holds.

Finally,

$$\begin{aligned}&\left| \int _{S_T} \int _{\Omega } \varrho _\varepsilon \textbf{u}_\varepsilon \cdot \mathcal {B} \left[ \varrho _\varepsilon ^\omega {\textrm{div}}_x\textbf{u}_\varepsilon - \frac{1}{|\Omega |} \int _{\Omega } \varrho _\varepsilon ^\omega {\textrm{div}}_x\textbf{u}_\varepsilon \ \,{\textrm{d}} {x} \right] \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \right| \\ {}&\quad \le \int _{S_T} \Vert \varrho _\varepsilon \Vert _{L^\gamma (\Omega )} \Vert \textbf{u}_\varepsilon \Vert _{L^6(\Omega ; R^3)} \left\| \mathcal {B} \left[ \varrho _\varepsilon ^\omega {\textrm{div}}_x\textbf{u}_\varepsilon - \frac{1}{|\Omega |} \int _{\Omega } \varrho _\varepsilon ^\omega {\textrm{div}}_x\textbf{u}_\varepsilon \ \,{\textrm{d}} {x} \right] \right\| _{L^q(\Omega ; R^3)} \ \,{{\textrm{d}}} t , \end{aligned}$$

where

$$\begin{aligned} \frac{1}{\gamma } + \frac{1}{6} + \frac{1}{q} = 1. \end{aligned}$$

Here,

$$\begin{aligned} \left\| \mathcal {B} \left[ \varrho _\varepsilon ^\omega {\textrm{div}}_x\textbf{u}_\varepsilon - \frac{1}{|\Omega |} \int _{\Omega } \varrho _\varepsilon ^\omega {\textrm{div}}_x\textbf{u}_\varepsilon \ \,{\textrm{d}} {x} \right] \right\| _{L^q(\Omega ; R^3)} {\mathop {\sim }\limits ^{<}}\Vert \varrho _\varepsilon ^\omega {\textrm{div}}_x\textbf{u}_\varepsilon \Vert _{L^r(\Omega ; R^3)},\ q = \frac{3 r}{3 - r}, \end{aligned}$$

and

$$\begin{aligned} \Vert \varrho _\varepsilon ^\omega {\textrm{div}}_x\textbf{u}_\varepsilon \Vert _{L^r(\Omega ; R^3)} \le \Vert \textbf{u}_\varepsilon \Vert _{W^{1,2}_0(\Omega ; R^3)} \Vert \varrho _\varepsilon ^\omega \Vert _{L^p(\Omega )},\ \text{ with }\ \frac{1}{2} + \frac{1}{p} = \frac{1}{r}. \end{aligned}$$

Thus using (4.14) we may infer that

$$\begin{aligned}&\left| \int _{S_T} \int _{\Omega } \varrho _\varepsilon \textbf{u}_\varepsilon \cdot \mathcal {B} \left[ \varrho _\varepsilon ^\omega {\textrm{div}}_x\textbf{u}_\varepsilon - \frac{1}{|\Omega |} \int _{\Omega } \varrho _\varepsilon ^\omega {\textrm{div}}_x\textbf{u}_\varepsilon \ \,{\textrm{d}} {x} \right] \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \right| \\ {}&\quad \le c(M_0) \sup _{t \in S_T} \Vert \varrho _\varepsilon \Vert _{L^{\frac{5}{3}} (\Omega )} \int _{S_T} \Vert \textbf{u}_\varepsilon \Vert ^2_{W^{1,2}_0(\Omega ; R^3)} \ \,{{\textrm{d}}} t . \end{aligned}$$

Going back to (4.15) and summarizing the previous estimates we conclude

$$\begin{aligned} \int _{S_T}&\int _{\Omega } \varrho _\varepsilon ^{\frac{5}{3} + \omega } \ \,{\textrm{d}} {x} \,{{\textrm{d}}} t \le c(M_0) \left( 1 + \int _{S_T} \int _{\Omega } \vartheta _\varepsilon ^4 \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \right. \nonumber \\&+ \sup _{t \in S_T} \Vert \varrho _\varepsilon \Vert _{L^{\frac{5}{3}} (\Omega )} \int _{S_T} \Vert \textbf{u}_\varepsilon \Vert ^2_{W^{1,2}_0(\Omega ; R^3)} \ \,{{\textrm{d}}} t \nonumber \\&+ \left. \int _{S_T} \int _{\Omega } {\mathbb {S}}(\vartheta _\varepsilon , {\mathbb {D}}_x\textbf{u}_\varepsilon ) : \nabla _x\mathcal {B} \left[ \varrho _\varepsilon ^\omega - \frac{1}{|\Omega |} \int _{\Omega } \varrho _\varepsilon ^\omega \ \,{\textrm{d}} {x} \right] \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \right) ,\ \text{ where }\ \omega = \frac{1}{15}. \end{aligned}$$
(4.16)

The last step is estimating

$$\begin{aligned}&\int _{\Omega } {\mathbb {S}} (\vartheta _\varepsilon , {\mathbb {D}}_x\textbf{u}_\varepsilon ) : \nabla _x\mathcal {B} \left[ \varrho _\varepsilon ^\omega - \frac{1}{|\Omega |} \int _{\Omega } \varrho ^\omega \ \,{\textrm{d}} {x} \right] \ \,{\textrm{d}} {x} \\&\quad \le (1 + \Vert \vartheta _\varepsilon \Vert _{L^4(\Omega )} ) \Vert \textbf{u}_\varepsilon \Vert _{W^{1,2}_0(\Omega ; R^3)} \left\| \nabla _x\mathcal {B} \left[ \varrho _\varepsilon ^\omega - \frac{1}{|\Omega |} \int _{\Omega } \varrho _\varepsilon ^\omega \ \,{\textrm{d}} {x} \right] \right\| _{L^4(\Omega ; R^3)} \\&\quad \le c(M) (1 + \Vert \vartheta _\varepsilon \Vert _{L^4(\Omega )} ) \Vert \textbf{u}_\varepsilon \Vert _{W^{1,2}_0(\Omega ; R^3)}. \end{aligned}$$

We therefore conclude the pressure estimates:

$$\begin{aligned} \int _{S_T}&\int _{\Omega } \varrho _\varepsilon ^{\frac{5}{3} + \omega } \ \,{\textrm{d}} {x} \,{{\textrm{d}}} t \le c(M_0) \left[ 1 + \int _{S_T} \int _{\Omega } \vartheta _\varepsilon ^4 \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \right. \nonumber \\&+ \left. \left( 1 + \sup _{t \in S_T} \Vert \varrho _\varepsilon \Vert _{L^{\frac{5}{3}} (\Omega )} \right) \int _{S_T} \Vert \textbf{u}_\varepsilon \Vert ^2_{W^{1,2}_0(\Omega ; R^3)} \ \,{{\textrm{d}}} t \right] ,\ \omega = \frac{1}{15}. \end{aligned}$$
(4.17)

4.4 Uniform bounds for \(\varepsilon \rightarrow 0\)

As \(\beta > 6\), we deduce from the inequalities (4.9), (4.17) that

$$\begin{aligned} \int _{S_T} \int _{\Omega } \vartheta _\varepsilon ^4 \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t&{\mathop {\sim }\limits ^{<}}\left( 1 + \int _{S_T} \Vert \vartheta _\varepsilon ^{\frac{\beta }{2}} \Vert ^2_{W^{1,2}(\Omega )} \ \,{{\textrm{d}}} t \right)&{\mathop {\sim }\limits ^{<}}\left( 1 + \int _{S_T} \int _{\Omega } \varrho _\varepsilon |\textbf{u}_\varepsilon | \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \right) \end{aligned}$$

provided we fix \(r = 1\) in (4.9). Furthermore,

$$\begin{aligned} \int _{S_T} \int _{\Omega } \varrho _\varepsilon |\textbf{u}_\varepsilon | \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t&\le \frac{1}{2} \int _{S_T} \int _{\Omega } \varrho _\varepsilon \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t + \frac{1}{2} \int _{S_T} \int _{\Omega } \varrho _\varepsilon |\textbf{u}_\varepsilon |^2 \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \\&\le \frac{1}{2}T M_0 + \frac{1}{2} \sup _{t \in S_T} \Vert \varrho _\varepsilon \Vert _{L^{\frac{5}{3}}(\Omega )} \int _{S_T} \Vert \textbf{u}_\varepsilon \Vert ^2_{L^5(\Omega ; R^3)} \ \,{{\textrm{d}}} t \\&\le c(M_0) \left( 1 + \sup _{t \in S_T} \Vert \varrho _\varepsilon \Vert _{L^{\frac{5}{3}}(\Omega )} \int _{S_T} \Vert \textbf{u}_\varepsilon \Vert ^2_{W^{1,2}_0(\Omega ; R^3)} \ \,{{\textrm{d}}} t \right) . \end{aligned}$$

Consequently, inequality (4.17) reduces to

$$\begin{aligned}&\int _{S_T} \int _{\Omega } \varrho _\varepsilon ^{\frac{5}{3} + \omega } \ \,{\textrm{d}} {x} \,{{\textrm{d}}} t \le c(M_0) \left[ 1 + \left( 1 + \sup _{t \in S_T} \Vert \varrho _\varepsilon \Vert _{L^{\frac{5}{3}} (\Omega )} \right) \int _{S_T} \Vert \textbf{u}_\varepsilon \Vert ^2_{W^{1,2}(\Omega ; R^3)} \ \,{{\textrm{d}}} t \right] ,\nonumber \\ {}&\omega = \frac{1}{15}. \end{aligned}$$
(4.18)

Next, going back to (4.9) we get

$$\begin{aligned} \int _{S_T} \Vert \textbf{u}_\varepsilon \Vert ^2_{W^{1,2}_0(\Omega ; R^3)} \ \,{{\textrm{d}}} t {\mathop {\sim }\limits ^{<}}\left( \mathcal {S}(r) \int _{S_T} \int _{\Omega } (\varrho _\varepsilon |\textbf{u}_\varepsilon | +\varrho _\varepsilon ) \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t + \Lambda (r) \right) \end{aligned}$$

where, by means of the standard Sobolev embedding theorem,

$$\begin{aligned}{} & {} \int _{\Omega } \varrho _\varepsilon |\textbf{u}_\varepsilon | \ \,{\textrm{d}} {x} \le \Vert \sqrt{\varrho _\varepsilon } \Vert _{L^2(\Omega )} \Vert \sqrt{\varrho _\varepsilon } \Vert _{L^3(\Omega )} \Vert \textbf{u}_\varepsilon \Vert _{L^6(\Omega ; R^3)}\\{} & {} \quad \le c(M_0) \Vert \sqrt{\varrho _\varepsilon } \Vert _{L^3(\Omega )} \Vert \textbf{u}_\varepsilon \Vert _{W^{1,2}_0(\Omega ; R^3)}. \end{aligned}$$

Consequently,

$$\begin{aligned} \int _{S_T} \Vert \textbf{u}_\varepsilon \Vert ^2_{W^{1,2}_0(\Omega ; R^3)} \ \,{{\textrm{d}}} t {\mathop {\sim }\limits ^{<}}\left( \mathcal {S}(r) \int _{S_T} \Vert \varrho _\varepsilon \Vert _{L^{\frac{3}{2}}(\Omega ) } \ \,{{\textrm{d}}} t + \Lambda (r) \right) . \end{aligned}$$
(4.19)

Now, introducing the total energy of the system,

$$\begin{aligned} E(\varrho , \vartheta , \textbf{u}) = \frac{1}{2} \varrho |\textbf{u}|^2 + \varrho e(\varrho , \vartheta ) \end{aligned}$$

we first observe that

$$\begin{aligned} \sup _{t \in S_T} \int _{\Omega } E(\varrho _\varepsilon , \vartheta _\varepsilon , \textbf{u}_\varepsilon ) \ \,{\textrm{d}} {x} {\mathop {\sim }\limits ^{<}}\left( 1 + \int _{S_T} \int _{\Omega } E(\varrho _\varepsilon , \vartheta _\varepsilon , \textbf{u}_\varepsilon ) \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \right) . \end{aligned}$$
(4.20)

The estimate (4.20) follows from the mean value theorem and the ballistic energy inequality (3.12). Indeed, in view of the uniform bounds established in Sect. 4.2, we first deduce (4.20) for the ballistic energy

$$\begin{aligned} E(\varrho _\varepsilon , \vartheta _\varepsilon , \textbf{u}_\varepsilon ) - \vartheta _B\varrho _\varepsilon s(\varrho _\varepsilon , \vartheta _\varepsilon ), \end{aligned}$$

and then use (2.9) to observe that the entropy part \(\vartheta _B \varrho _\varepsilon s(\varrho _\varepsilon , \vartheta _\varepsilon )\) is a lower order perturbation.

Now, we estimate the kinetic energy using (4.19),

$$\begin{aligned}&\int _{S_T} \int _{\Omega } \varrho _\varepsilon |\textbf{u}_\varepsilon |^2 \ \,{\textrm{d}} {x} \,{{\textrm{d}}} t \\ {}&\quad \le \sup _{t \in S_T} \Vert \varrho _\varepsilon \Vert _{L^{\frac{3}{2}}(\Omega )} \int _{S_T} \Vert \textbf{u}_\varepsilon \Vert ^2_{L^6(\Omega ; R^3)} \ \,{{\textrm{d}}} t \\ {}&\quad \le c \sup _{t \in S_T} \Vert \varrho _\varepsilon \Vert _{L^{\frac{3}{2}}(\Omega )} \int _{\tau }^{\tau + 1} \Vert \textbf{u}_\varepsilon \Vert ^2_{W^{1,2}_0(\Omega ; R^3)} \,{{\textrm{d}}} t \\&\quad {\mathop {\sim }\limits ^{<}}\Lambda (r) \sup _{t \in S_T} \Vert \varrho _\varepsilon \Vert _{L^{\frac{3}{2}}(\Omega )} + \mathcal {S}(r) \sup _{t \in S_T} \Vert \varrho _\varepsilon \Vert _{L^{\frac{3}{2}}(\Omega )}\int _{S_T} \Vert \varrho _\varepsilon \Vert _{L^{\frac{3}{2}}(\Omega )} \ \,{{\textrm{d}}} t . \end{aligned}$$

In addition, by interpolation,

$$\begin{aligned} \Vert \varrho _\varepsilon \Vert _{L^{\frac{3}{2}}(\Omega )} \le \Vert \varrho _\varepsilon \Vert _{L^{\frac{5}{3}}(\Omega )}^{\frac{5}{6}} \Vert \varrho _\varepsilon \Vert _{L^1(\Omega )}^{\frac{1}{6}}. \end{aligned}$$
(4.21)

Consequently,

$$\begin{aligned} \int _{S_T}&\int _{\Omega } \varrho _\varepsilon |\textbf{u}_\varepsilon |^2 \ \,{\textrm{d}} {x} \,{{\textrm{d}}} t \le \nonumber \\&{\mathop {\sim }\limits ^{<}}c(M_0) \Lambda (r) \sup _{t \in S_T} \Vert \varrho _\varepsilon \Vert _{L^{\frac{5}{3}}(\Omega )}^{\frac{5}{6}} + \mathcal {S}(r) \sup _{t \in S_T} \Vert \varrho _\varepsilon \Vert _{L^{\frac{5}{3}}(\Omega )}^{\frac{5}{6}} \int _{S_T} \Vert \varrho _\varepsilon \Vert _{L^{\frac{5}{3}}(\Omega )}^{\frac{5}{6}} \ \,{{\textrm{d}}} t . \end{aligned}$$
(4.22)

Combining (4.18), (4.19), (4.21) we get

$$\begin{aligned}&\int _{S_T} \int _{\Omega } \varrho _\varepsilon ^{\frac{5}{3} + \omega } \ \,{\textrm{d}} {x} \,{{\textrm{d}}} t \nonumber \\&\quad \le c(M_0) \left[ 1 + \left( 1 + \sup _{t \in S_T} \Vert \varrho _\varepsilon \Vert _{L^{\frac{5}{3}} (\Omega )} \right) \int _{S_T} \Vert \textbf{u}_\varepsilon \Vert ^2_{W^{1,2}_0(\Omega ; R^3)} \ \,{{\textrm{d}}} t \right] \nonumber \\&\quad \le c(M_0) \left[ 1 + \left( 1 + \sup _{t \in S_T} \Vert \varrho _\varepsilon \Vert _{L^{\frac{5}{3}} (\Omega )} \right) \left( \mathcal {S}(r) \int _{S_T} \Vert \varrho _\varepsilon \Vert _{L^{\frac{3}{2}}(\Omega ) } \ \,{{\textrm{d}}} t + \Lambda (r) \right) \right] \nonumber \\&\quad \le c(M_0) \left[ 1 + \left( 1 + \sup _{t \in S_T} \Vert \varrho _\varepsilon \Vert _{L^{\frac{5}{3}} (\Omega )} \right) \left( \mathcal {S}(r) \int _{S_T} \Vert \varrho _\varepsilon \Vert _{L^{\frac{5}{3}}(\Omega ) }^{\frac{5}{6}} \ \,{{\textrm{d}}} t + \Lambda (r) \right) \right] . \end{aligned}$$
(4.23)

Interpolating \(L^1\) and \(L^{\frac{5}{3} + \omega }\) and using boundedness of the total mass we have

$$\begin{aligned} \int _{S_T} \int _{\Omega } \varrho _\varepsilon ^{\frac{5}{3}} \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \le c(M_0) \left( \int _{S_T} \int _{\Omega } \varrho _\varepsilon ^{\frac{5}{3} + \omega } \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \right) ^{\frac{10}{11}} \ \text{ provided }\ \omega = \frac{1}{15}.\qquad \end{aligned}$$
(4.24)

Thus summing up (4.20)–(4.24) we may infer that

$$\begin{aligned}&{}(4.20) \Rightarrow \ \sup _{t \in S_T} \int _{\Omega } E(\varrho _\varepsilon , \vartheta _\varepsilon , \textbf{u}_\varepsilon ) \ \,{\textrm{d}} {x} {\mathop {\sim }\limits ^{<}}\left( 1 + \int _{S_T} \int _{\Omega } E(\varrho _\varepsilon , \vartheta _\varepsilon , \textbf{u}_\varepsilon ) \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \right) \nonumber \\&\quad {\mathop {\sim }\limits ^{<}}\left[ 1 + \int _{S_T} \int _{\Omega } \left( \Vert \textbf{u}_\varepsilon \Vert _{W^{1,2}_0(\Omega ; R^d)}^2 + \left\| \vartheta _\varepsilon ^{\frac{\beta }{2}} \right\| _{W^{1,2}(\Omega )}^2 + \left\| \log (\vartheta _\varepsilon ) \right\| _{W^{1,2}(\Omega )}^2 \right) \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \right. \nonumber \\&\qquad + \left. \int _{S_T} \int _{\Omega } \varrho _\varepsilon |\textbf{u}_\varepsilon |^2 \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t + \int _{S_T} \int _{\Omega } \varrho _\varepsilon ^{\frac{5}{3}} \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \right] \nonumber \\&\qquad (4.9) \Rightarrow {\mathop {\sim }\limits ^{<}}\left[ 1 + \int _{S_T} \int _{\Omega } \varrho _\varepsilon |\textbf{u}_\varepsilon |^2 \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t + \int _{S_T} \int _{\Omega } \varrho _\varepsilon ^{\frac{5}{3}} \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \right] \nonumber \\&\qquad (4.22) \Rightarrow {\mathop {\sim }\limits ^{<}}\left[ 1 + \int _{S_T} \int _{\Omega } \varrho _\varepsilon ^{\frac{5}{3}} \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \right. \nonumber \\&\qquad + \left. \Lambda (r) \left( \sup _{t \in S_T} \int _{\Omega } E(\varrho _\varepsilon , \vartheta _\varepsilon , \textbf{u}_\varepsilon ) \ \,{\textrm{d}} {x} \right) ^{\frac{1}{2}} + \mathcal {S}(r) \left( \sup _{t \in S_T} \int _{\Omega } E(\varrho _\varepsilon , \vartheta _\varepsilon , \textbf{u}_\varepsilon ) \ \,{\textrm{d}} {x} \right) \right] \nonumber \\&\qquad (4.24) \Rightarrow \le c(M_0) \left[ 1 + \left( \int _{S_T} \int _{\Omega } \varrho _\varepsilon ^{\frac{5}{3} + \omega } \ \,{\textrm{d}} {x} \ \,{{\textrm{d}}} t \right) ^{\frac{10}{11}} \right. \nonumber \\&\qquad + \left. \Lambda (r) \left( \sup _{t \in S_T} \int _{\Omega } E(\varrho _\varepsilon , \vartheta _\varepsilon , \textbf{u}_\varepsilon ) \ \,{\textrm{d}} {x} \right) ^{\frac{1}{2}} + \mathcal {S}(r) \left( \sup _{t \in S_T} \int _{\Omega } E(\varrho _\varepsilon , \vartheta _\varepsilon , \textbf{u}_\varepsilon ) \ \,{\textrm{d}} {x} \right) \right] \nonumber \\&\qquad (4.23) \Rightarrow \le c(M_0) \left[ \Lambda (r) + \mathcal {S}^{\frac{10}{11}}(r) \left( \sup _{t \in S_T} \int _{\Omega } E(\varrho _\varepsilon , \vartheta _\varepsilon , \textbf{u}_\varepsilon ) \ \,{\textrm{d}} {x} \right) \right. \nonumber \\&\qquad + \left. \Lambda (r) \left( \sup _{t \in S_T} \int _{\Omega } E(\varrho _\varepsilon , \vartheta _\varepsilon , \textbf{u}_\varepsilon ) \ \,{\textrm{d}} {x} \right) ^{\frac{1}{2}} + \mathcal {S}(r) \left( \sup _{t \in S_T} \int _{\Omega } E(\varrho _\varepsilon , \vartheta _\varepsilon , \textbf{u}_\varepsilon ) \ \,{\textrm{d}} {x} \right) \right] . \end{aligned}$$
(4.25)

As \(\mathcal {S}(r) \rightarrow 0\) as \(r \rightarrow \infty \), we fix \(r > 0\) large enough to deduce from (4.25) the desired energy bound

$$\begin{aligned} \sup _{t \in S_T} \int _{\Omega } E(\varrho _\varepsilon , \vartheta _\varepsilon , \textbf{u}_\varepsilon ) \ \,{\textrm{d}} {x} \le c(M_0). \end{aligned}$$
(4.26)

5 Convergence

Our ultimate goal is to perform the limit in the sequence of approximate solutions \((\varrho _\varepsilon , \vartheta _\varepsilon , \textbf{u}_\varepsilon )_{\varepsilon > 0}\) to obtain the existence of the time–periodic solution claimed in Theorem 2.2. With the energy estimate (4.26) at hand, this is nowadays well understood routine matter. Indeed the test functions used in the entropy inequality (2.16) are compactly supported thus unaffected by the boundary integral in its approximate counterpart (3.6). Similarly, the approximate ballistic energy (3.12) is in fact stronger than (2.17) due to the penalization

$$\begin{aligned} \frac{1}{\varepsilon } \int _{S_T} \psi \int _{\partial \Omega } \frac{ |\vartheta _\varepsilon - \vartheta _B|^{k+2} }{\vartheta _\varepsilon } {\textrm{d}}\sigma _x \ \,{{\textrm{d}}} t {\mathop {\sim }\limits ^{<}}1,\ \psi \ge 0. \end{aligned}$$
(5.1)

In particular, for \(\psi = 1\), the above inequality together with the (3.12) yield

$$\begin{aligned} \vartheta _\varepsilon \rightarrow \vartheta \ \text{ weakly } \text{ in }\ L^2(0,T; W^{1,2}(\Omega ; R^d)) \end{aligned}$$

with the limit trace \(\vartheta |_{\partial \Omega } = \vartheta _B\) as required in Theorem 2.2.

Consequently, the proof of convergence is exactly the same as in the existence theory elaborated in [7] with the exception of the strong convergence of the density, the “initial” value of which is unspecified in the periodic setting. Fortunately, the compactness arguments based on Lions’ identity and boundedness of the oscillation defect measure can be modified to accommodate the time periodic setting exactly as in [10, Section 9.3]. Thus the proof of Theorem 2.2 can be completed.

6 Concluding remarks

In comparison with [10], the available a priori bounds do not allow to handle a general driving force \(\varrho \textbf{g}\) in the momentum equation. Although the potential case \(\textbf{g} = \nabla _xG\) is physically relevant, more general (non–potential) forces occur when the fluid is stirred up by the motion of the container. A detailed inspection of the arguments in Sect. 4.3 reveals that they could be considerably improved in the case \(d=2\) due to the Sobolev embedding \(W^{1,2} \subset L^q\) for any finite q. Similar improvement may also be expected in the case the total mass \(M_0\) is small, cf. Wang and Wang [26]. We therefore strongly conjecture that the present result can be extended to a general driving force \(\textbf{g}\) provided

  • either \(d=2\),

  • or

    $$\begin{aligned} M_0 = \int _{\Omega } \varrho \ \,{\textrm{d}} {x} \end{aligned}$$

    is small enough with respect to the amplitude of \(\textbf{g}\).

As potentiality of \(\textbf{g}\) was also used in the estimate (3.8) crucial for boundedness of the approximate sequence, the proof of the above conjecture would require a different kind of approximation scheme.

Finally, let us discuss briefly the possibility of extending the results to non–smooth spatial domains. In view of the Sobolev space theory, notably various embedding theorems, one is tempted to say that everything works well for domains with Lipschitz boundary. Indeed we believe that such an extension is possible, however, there are some technical difficulties to overcome in the construction of the approximate solutions, see e.g. Poul [22].