Existence Theory

Before formulating our main existence result, we present a concise list of hypotheses imposed on the data. To see their interpretation, the reader may consult Chap. 1 for the physical background and the relevant discussion.

(i) Initial data: The initial state of the system is determined through the choice of the quantities ϱ ₀, (ϱ u)₀, E ₀, and (ϱs)₀.

The initial density ϱ ₀ is a non-negative measurable function such that

The initial distribution of the momentum satisfies a compatibility condition

notably the total amount of the kinetic energy is finite, meaning,

The initial temperature is determined by a measurable function ϑ ₀ satisfying

Finally, we assume that he initial energy of the system is finite, specifically,

(ii) Source terms: For the sake of simplicity, we suppose that

(iii) Constitutive relations: The quantities p, e, and s are continuously differentiable functions for positive values of ϱ, ϑ satisfying Gibbs’ equation

In addition,

and

where, in accordance with hypothesis of thermodynamic stability (1.44), the molecular components satisfy

and

Furthermore,

and,

Finally, we suppose that there is a function P satisfying

and two positive constants \(0 <\underline{ Z} <\overline{Z}\) such that

where, in addition,

(iv) Transport coefficients: The viscosity coefficients μ, η are continuously differentiable functions of the absolute temperature ϑ, more precisely μ,  η ∈ C ¹[0, ∞), satisfying

The heat conductivity coefficient κ can be decomposed as

where κ _M ,  κ _R ∈ C ¹[0, ∞), and

In formulas (3.17)–(3.22), \(\underline{\mu }\), \(\overline{\mu }\), \(\overline{m}\), \(\overline{\eta }\), \(\underline{\kappa }_{R}\), \(\overline{\kappa }_{R}\), \(\underline{\kappa }_{M}\), \(\overline{\kappa }_{M}\) are positive constants and

Remark

Some of the above hypotheses, in particular those imposed on the thermodynamic functions, are rather technical and may seem awkward at first glance. The reader should always keep in mind the prototype example

which meets all the hypotheses stated above. Note that if a > 0 is small and P(Z) is close to a linear function for moderate values of Z, the above formula approaches the standard Boyle-Marriot law of a perfect gas.

The present hypotheses cover, in particular, the physically reasonable case when the constitutive law for the molecular pressure is that one of the monoatomic gas, meaning

for more details see Sect. 1.4.2.

Very roughly indeed, we can say that the pressure is regularized in the area where either ϱ or ϑ are close to zero. The radiation component p _R prevents the temperature field from oscillating in the vacuum zone where ϱ vanishes, while the superlinear growth of P for large arguments guarantees strong enough a priori estimates on the density ϱ in the “cold” regime ϑ ≈ 0.

The hypotheses on constitutive relations for the pressure, the internal energy and the entropy entail further restrictions imposed on the structural properties of the functions p, e, and s. Some of them have already been identified and used in Chap. 2 For reader’s convenience, they are recorded and studied in a systematic way in the text below.

1. (i)

   The first observation is that for (3.15), (3.16) to be compatible with the hypothesis of thermodynamic stability expressed through (3.10), (3.11), the function P must obey certain structural restrictions. In particular, relation (3.10) yields

   $$\displaystyle{P'(Z)> 0\ \mbox{ whenever}\ 0 <Z <\underline{ Z},\ \mbox{ or,}\ Z> \overline{Z},}$$

   which, together with (3.14), yields

   $$\displaystyle{ P'(Z)> 0\ \mbox{ for all}\ Z \geq 0, }$$

   (3.24)

   where P has been extended to be strictly increasing on the interval \([\underline{Z},\overline{Z}]\).

   Similarly, a direct inspection of (3.11), (3.15), (3.16) gives rise to

   $$\displaystyle{ 0 <\frac{3} {2} \frac{\frac{5} {3}P(Z) - ZP'(Z)} {Z}:= c_{v,M} <c,\;\ \mbox{ whenever}\ Z = \frac{\varrho } {\vartheta ^{3/2}} \geq \overline{Z}. }$$

   (3.25)

   In particular P(Z)∕Z ^5∕3 possesses a limit for Z → ∞, specifically, in accordance with (3.15), (3.16),

   $$\displaystyle{\lim _{\vartheta \rightarrow 0+}e_{M}(\varrho,\vartheta ) = \frac{3} {2}\lim _{\vartheta \rightarrow 0+}\frac{\vartheta ^{5/2}} {\varrho } P\Big( \frac{\varrho } {\vartheta ^{3/2}}\Big) = \frac{3} {2}\varrho ^{\frac{2} {3} }\lim _{Z\rightarrow \infty }\frac{P(Z)} {Z^{5/3}} \ \mbox{ for any fixed}\ \varrho> 0.}$$

   Moreover, in agreement with (3.12),

   $$\displaystyle{ \lim _{Z\rightarrow \infty }\frac{P(Z)} {Z^{5/3}} = p_{\infty }> 0, }$$

   (3.26)

   and

   $$\displaystyle{ \lim _{\vartheta \rightarrow 0+}e_{M}(\varrho,\vartheta ) =\underline{ e}_{M}(\varrho ) = \frac{3} {2}\varrho ^{2/3}p_{ \infty }. }$$

   (3.27)
2. (ii)

   By virtue of (3.11), the function ϑ ↦ e _M (ϱ, ϑ) is strictly increasing on the whole interval (0, ∞) for any fixed ϱ > 0. This fact, together with (3.9), (3.27), gives rise to the lower bound

   $$\displaystyle{ \varrho e(\varrho,\vartheta ) \geq \frac{3p_{\infty }} {2} \varrho ^{\frac{5} {3} } + a\vartheta ^{4}. }$$

   (3.28)

   On the other hand,

   $$\displaystyle{ e_{M}(\varrho,\vartheta ) =\underline{ e}_{M}(\varrho ) +\int _{ 0}^{\vartheta }\frac{\partial e_{M}} {\partial \vartheta } (\varrho,\tau )\,\mathrm{d}\tau, }$$

   (3.29)

   which, together with (3.11) and (3.27), yields

   $$\displaystyle{ 0 \leq e_{M}(\varrho,\vartheta ) \leq c(\varrho ^{\frac{2} {3} }+\vartheta ). }$$

   (3.30)

   Similarly, relation (3.24), together with (3.14)–(3.16), and (3.26), yield the following bounds on the molecular pressure p _M :

   $$\displaystyle{ \underline{c}\varrho \vartheta \leq p_{M}(\varrho,\vartheta ) \leq \overline{c}\varrho \vartheta \quad \mbox{ }\mbox{ if}\ \varrho <\overline{Z}\vartheta ^{\frac{3} {2} }, }$$

   (3.31)

   and

   $$\displaystyle{ \underline{c}\varrho ^{\frac{5} {3} } \leq p_{M}(\varrho,\vartheta ) \leq \overline{c}\left \{\begin{array}{c} \vartheta ^{\frac{5} {2} }\ \mbox{ if}\ \varrho <\overline{Z}\vartheta ^{\frac{3} {2} }\\ \\ \varrho ^{ \frac{5} {3} }\ \mbox{ if}\ \varrho> \overline{Z}\vartheta ^{\frac{3} {2} }.\end{array} \right \} }$$

   (3.32)

   Here, we have used the monotonicity of p _M in ϱ in order to control the behavior of the pressure in the region

   $$\displaystyle{\underline{Z}\vartheta ^{\frac{3} {2} } \leq \varrho \leq \overline{Z}\vartheta ^{\frac{3} {2} }.}$$

   Moreover, in accordance with (3.30), (3.32), it is easy to observe that

   $$\displaystyle{ e_{M},p_{M}\quad \mbox{ are bounded on bounded sets of }[0,\infty )^{2}. }$$

   (3.33)
3. (iii)

   In agreement with Gibbs’ relation (3.7), the specific entropy s can be written as

   $$\displaystyle{ s = s_{M} + s_{R},\quad \frac{\partial s_{M}} {\partial \vartheta } = \frac{1} {\vartheta } \frac{\partial e_{M}} {\partial \vartheta },\quad \varrho s_{R}(\varrho,\vartheta ) = \frac{4} {3}a\vartheta ^{3}, }$$

   (3.34)

   where the molecular component s _M satisfies

   $$\displaystyle{ s_{M}(\varrho,\vartheta ) = S(Z),\,Z = \frac{\varrho } {\vartheta ^{3/2}},\,S'(Z) = -\frac{3} {2} \frac{\frac{5} {3}P(Z) - ZP'(Z)} {Z^{2}} <0 }$$

   (3.35)

   in the degenerate area \(\varrho> \overline{Z}\vartheta ^{\frac{3} {2} }\). Note that the function S is determined up to an additive constant.

   On the other hand, due to (3.11), the function ϑ ↦ s _M (ϱ, ϑ) is increasing on (0, ∞) for any fixed ϑ. Accordingly,

   $$\displaystyle{ s_{M}(\varrho,\vartheta ) \leq \left \{\begin{array}{c} s_{M}(\varrho,1)\quad \mbox{ if }\vartheta \leq 1\\ \\ s_{M}(\varrho,1) +\int _{ 1}^{\vartheta }\frac{\partial s_{M}} {\partial \vartheta } (\varrho,\tau )\,\mathrm{d}\tau \leq s_{M}(\varrho,1) + c\log \vartheta \quad \mbox{ if }\vartheta> 1 \end{array} \right \}, }$$

   (3.36)

   where we have exploited (3.11) combined with (3.34) in order to control

   $$\displaystyle{ \left \vert \int _{1}^{\vartheta }\frac{\partial s_{M}} {\partial \vartheta } (\varrho,\tau )\ \mathrm{d}\tau \right \vert \leq c\vert \log \vartheta \vert \ \mbox{ for all}\ \vartheta> 0. }$$

   (3.37)

   Another application of Gibbs’ relation (3.7) yields

   $$\displaystyle{\frac{\partial s_{M}} {\partial \varrho } = -\frac{1} {\varrho ^{2}} \frac{\partial p_{M}} {\partial \vartheta },}$$

   see also (1.3); therefore

   $$\displaystyle{s_{M}(\varrho,1) = s_{M}(1,1) +\int _{ 1}^{\varrho }\frac{1} {\tau ^{2}} \frac{\partial p_{M}} {\partial \vartheta } (\tau,1)\,\mathrm{d}\tau.}$$

   By virtue of (3.15) and (3.25),

   $$\displaystyle{\frac{\partial p_{M}} {\partial \vartheta } (\rho,1) = \frac{5} {2}P(\varrho ) -\frac{3} {2}\varrho P'(\varrho ) \leq c\varrho \ \mbox{ for all}\ \varrho \in (0,\underline{Z}] \cup [\overline{Z},\infty ),}$$

   whereas

   $$\displaystyle{\vert \frac{\partial p_{M}} {\partial \vartheta } (\rho,1)\vert \quad \mbox{ is bounded in }[\underline{Z},\overline{Z}].}$$

   Consequently,

   $$\displaystyle{ \vert s_{M}(\varrho,1)\vert \leq c(1 + \vert \log \varrho \vert )\ \mbox{ for all}\ \varrho \in (0,\infty ). }$$

   (3.38)

   Writing

   $$\displaystyle{s_{M}(\varrho,\vartheta ) = s_{M}(\varrho,1) +\int _{ 1}^{\varrho }\frac{\partial s_{M}} {\partial \vartheta } (\varrho,\tau )\ \mathrm{d}\tau }$$

   and resuming the previous estimates, we conclude that

   $$\displaystyle{ \vert s_{M}(\varrho,\vartheta )\vert \leq c(1 + \vert \log \varrho \vert + \vert \log \vartheta \vert )\ \mbox{ for all}\ \varrho,\vartheta> 0. }$$

   (3.39)
4. (iv)

   It follows from (3.35) that

   $$\displaystyle{ \begin{array}{c} \lim _{Z\rightarrow \infty }S(Z) = s_{\infty } = \left \{\begin{array}{c} -\infty \\ \\ 0\end{array} \right \};\\ \\ \mbox{ whence}\\ \\ \lim _{\vartheta \rightarrow 0+}s_{M}(\varrho,\vartheta ) = s_{\infty }\ \mbox{ for any fixed}\ \varrho> 0.\end{array} }$$

   (3.40)

   where, in the latter case, we have fixed the free additive constant in the definition of S in (3.35) to obtain s _∞ = 0.

5. (v)

   Finally, as a direct consequence of (3.15),

   $$\displaystyle{\frac{\partial p_{M}} {\partial \varrho } (\varrho,\vartheta ) = \vartheta P'\left ( \frac{\varrho } {\vartheta ^{\frac{3} {2} }} \right )\mbox{ if}\ \varrho <\underline{ Z}\vartheta ^{\frac{3} {2} },\ \mbox{ or,}\ \varrho> \overline{Z}\vartheta ^{\frac{3} {2} },}$$

   where, by virtue of (3.24)–(3.26),

   $$\displaystyle{ P'(Z) \geq c(1 + Z^{\frac{2} {3} }),\ c> 0,\ \mbox{ for all}\ Z \geq 0. }$$

   (3.41)

   Thus we can write

   $$\displaystyle{p_{M}(\varrho,\vartheta ) = \vartheta ^{\frac{5} {2} }P\left ( \frac{\varrho } {\vartheta ^{\frac{3} {2} }} \right ) + p_{b}(\varrho,\vartheta ),}$$

   with

   $$\displaystyle{p_{b}(\varrho,\vartheta ) = p_{M}(\varrho,\vartheta ) -\vartheta ^{\frac{5} {2} }P\left ( \frac{\varrho } {\vartheta ^{\frac{3} {2} }} \right ).}$$

   In accordance with (3.15), (3.32), we have

   $$\displaystyle{ \vert p_{b}(\varrho,\vartheta )\vert \leq c(1 + \vartheta ^{\frac{5} {2} }). }$$

   (3.42)

   Finally, we conclude with help of (3.41) that there exists d > 0 such that

   $$\displaystyle{ p_{M}(\varrho,\vartheta ) = d\varrho ^{\frac{5} {3} } + p_{m}(\varrho,\vartheta ) + p_{b}(\varrho,\vartheta ), }$$

   (3.43)

   where

   $$\displaystyle{ \frac{\partial p_{m}} {\partial \varrho } (\varrho,\vartheta )> 0\ \mbox{ for all}\ \varrho,\vartheta> 0. }$$

   (3.44)

Having collected all the preliminary material, we are in a position to formulate our main existence result concerning the weak solutions of the Navier-Stokes-Fourier system.

■      Global Existence for the Navier-Stokes-Fourier System:

Theorem 3.1

Let \(\Omega \subset \mathbb{R}^{3}\) be a bounded domain of class C ^2,ν , ν ∈ (0, 1). Assume that

• the data ϱ ₀ , (ϱ u)₀ , E ₀ , (ϱs)₀ satisfy ( 3.1 )–( 3.5 );

• the source terms f , \(\mathcal{Q}\) are given by ( 3.6 );

• the thermodynamic functions p, e, s, and the transport coefficients μ, η, κ obey the structural hypotheses ( 3.7 )–( 3.23 ).

Then for any T > 0 the Navier-Stokes-Fourier system admits a weak solution {ϱ, u, ϑ} on \((0,T) \times \Omega\) in the sense specified in Sect.  2.1 . More precisely, {ϱ, u, ϑ} satisfy relations ( 2.2 )–(2.6), (2.9)–(2.17), (2.22)–(2.25), (2.27)–(2.32), with (2.35)–(2.37).

The complete proof of Theorem 3.1 presented in the remaining part of this chapter is tedious, rather technical, consisting in four steps:

• The momentum equation (2.9) is replaced by a Faedo-Galerkin approximation, the equation of continuity (2.2) is supplemented with an artificial viscosity term, and the entropy production equation (2.27) is replaced by the balance of internal energy. The approximate solutions are obtained by help of the Schauder fixed point theorem, first locally in time, and then extended on the full interval (0, T) by means of suitable uniform estimates.

• Performing the limit in the Faedo-Galerkin approximation scheme we recover the momentum equation supplemented with an artificial pressure term. Simultaneously, the balance of internal energy is converted to the entropy production equation (2.27), together with the total energy balance (2.22) containing some extra terms depending on small parameters.

• We pass to the limit in the regularized equation of continuity sending the artificial viscosity terms to zero.

• Finally, the proof of Theorem 3.1 is completed letting the artificial pressure term go to zero.

3.3.1 Approximation Scheme

1. (i)

   The equation of continuity (2.2) is regularized by means of an artificial viscosity term:

   $$\displaystyle{ \partial _{t}\varrho + \mathrm{div}_{x}(\varrho \mathbf{u}) = \varepsilon \Delta \varrho \ \mbox{ in}\ (0,T) \times \Omega, }$$

   (3.45)

   and supplemented with the homogeneous Neumann boundary condition

   $$\displaystyle{ \nabla _{x}\varrho \cdot \mathbf{n}\vert _{\partial \Omega } = 0, }$$

   (3.46)

   and the initial condition

   $$\displaystyle{ \varrho (0,\cdot ) =\varrho _{0,\delta }, }$$

   (3.47)

   where

   $$\displaystyle{ \varrho _{0,\delta } \in C^{2,\nu }(\overline{\Omega }),\ \inf _{ x\in \Omega }\varrho _{0,\delta }(x)> 0,\ \nabla _{x}\varrho _{0,\delta } \cdot \mathbf{n}\vert _{\partial \Omega } = 0. }$$

   (3.48)
2. (ii)

   The momentum balance expressed through the integral identity (2.9) is replaced by a Faedo-Galerkin approximation:

   $$\displaystyle{ \int _{0}^{T}\int _{ \Omega }\Big(\varrho \mathbf{u} \cdot \partial _{t}\boldsymbol{\varphi } +\varrho [\mathbf{u} \otimes \mathbf{u}]: \nabla _{x}\boldsymbol{\varphi } +\Big (\,p(\varrho,\vartheta ) +\delta (\varrho ^{\Gamma } +\varrho ^{2})\Big)\mathrm{div}_{ x}\boldsymbol{\varphi }\Big)\ \mathrm{d}x\ \mathrm{d}t }$$

   (3.49)
   $$\displaystyle{=\int _{ 0}^{T}\int _{ \Omega }\Big(\varepsilon (\nabla _{x}\varrho \nabla _{x}\mathbf{u}) \cdot \boldsymbol{\varphi } +\mathbb{S}_{\delta }: \nabla _{x}\boldsymbol{\varphi } -\varrho \mathbf{f}_{\delta }\cdot \boldsymbol{\varphi }\Big)\ \mathrm{d}x\ \mathrm{d}t -\int _{\Omega }(\varrho \mathbf{u})_{0}\cdot \boldsymbol{\varphi }\ \mathrm{d}x,}$$

   to be satisfied for any test function \(\boldsymbol{\varphi }\in C_{c}^{1}([0,T);X_{n})\), where

   $$\displaystyle{ X_{n} \subset C^{2,\nu }(\overline{\Omega }; \mathbb{R}^{3}) \subset L^{2}(\Omega; \mathbb{R}^{3}) }$$

   (3.50)

   is a finite-dimensional vector space of functions satisfying either

   $$\displaystyle{ \boldsymbol{\varphi }\cdot \mathbf{n}\vert _{\partial \Omega } = 0\ \mbox{ in the case of the complete slip boundary conditions,} }$$

   (3.51)

   or

   $$\displaystyle{ \boldsymbol{\varphi }\vert _{\partial \Omega } = 0\ \mbox{ in the case of the no-slip boundary conditions.} }$$

   (3.52)

   The space X _n is endowed with the Hilbert structure induced by the scalar product of the Lebesgue space \(L^{2}(\Omega; \mathbb{R}^{3})\).

   Furthermore, we set

   $$\displaystyle{ \mathbb{S}_{\delta } = \mathbb{S}_{\delta }(\vartheta,\nabla _{x}\mathbf{u})\ = (\mu (\vartheta )+\delta \vartheta )\Big(\nabla _{x}\mathbf{u} + \nabla _{x}^{T}\mathbf{u} -\frac{2} {3}\mathrm{div}_{x}\mathbf{u}\ \mathbb{I}\Big) +\eta (\vartheta )\mathrm{div}_{x}\mathbf{u}\ \mathbb{I}, }$$

   (3.53)

   while the function

   $$\displaystyle{ \mathbf{f}_{\delta } \in C^{1}([0,T] \times \overline{\Omega }; \mathbb{R}^{3}) }$$

   (3.54)

   is a suitable approximation of the driving force f.

3. (iii)

   Instead of the entropy balance (2.27), we consider a modified internal energy equation in the form:

   $$\displaystyle{ \partial _{t}(\varrho e_{\delta }(\varrho,\vartheta )) + \mathrm{div}_{x}(\varrho e_{\delta }(\varrho,\vartheta )\mathbf{u}) -\mathrm{div}_{x}\nabla _{x}\mathcal{K}_{\delta }(\vartheta ) }$$

   (3.55)
   $$\displaystyle{= \mathbb{S}_{\delta }(\vartheta,\nabla _{x}\mathbf{u}): \nabla _{x}\mathbf{u} - p(\varrho,\vartheta )\mathrm{div}_{x}\mathbf{u} +\varrho \mathcal{Q}_{\delta } + \varepsilon \delta (\Gamma \varrho ^{\Gamma -2} + 2)\vert \nabla _{ x}\varrho \vert ^{2} +\delta \frac{1} {\vartheta ^{2}} -\varepsilon \vartheta ^{5},}$$

   supplemented with the Neumann boundary condition

   $$\displaystyle{ \nabla _{x}\vartheta \cdot \mathbf{n}\vert _{\partial \Omega } = 0, }$$

   (3.56)

   and the initial condition

   $$\displaystyle{ \vartheta (0,\cdot ) =\vartheta _{0,\delta }, }$$

   (3.57)
   $$\displaystyle{ \vartheta _{0,\delta } \in W^{1,2}(\Omega ) \cap L^{\infty }(\Omega ),\ \mathrm{ess}\inf _{ x\in \Omega }\vartheta _{0,\delta }(x)> 0. }$$

   (3.58)

   Here

   $$\displaystyle{ e_{\delta }(\varrho,\vartheta ) = e_{M,\delta }(\varrho,\vartheta ) + a\vartheta ^{4},\quad e_{ M,\delta }(\varrho,\vartheta ) = e_{M}(\varrho,\vartheta )+\delta \vartheta, }$$

   (3.59)
   $$\displaystyle{\mathcal{K}_{\delta }(\vartheta ) =\int _{ 1}^{\vartheta }\kappa _{ \delta }(z)\ \mathrm{d}z,\ \kappa _{\delta }(\vartheta ) =\kappa _{M}(\vartheta ) +\kappa _{R}(\vartheta ) +\delta \Big (\vartheta ^{\Gamma } + \frac{1} {\vartheta } \Big),}$$

   and

   $$\displaystyle{ \mathcal{Q}_{\delta }\geq 0,\ \mathcal{Q}_{\delta }\in C^{1}([0,T] \times \overline{\Omega }). }$$

   (3.60)

In problem (3.45)–(3.60), the quantities ɛ, δ are small positive parameters, while \(\Gamma> 0\) is a sufficiently large fixed number. The meaning of the extra terms will become clear in the course of the proof. Loosely speaking, the ɛ-dependent quantities provide more regularity of the approximate solutions modifying the type of the field equations, while the δ-dependent quantities prevent concentrations yielding better estimates on the amplitude of the approximate solutions. For technical reasons, the limit passage must be split up in two steps letting first ɛ → 0 and then δ → 0.

We claim the following result concerning solvability of the approximate problem (3.45)–(3.60).

■      Global Existence for the Approximate System:

Proposition 3.1

Let ɛ, δ be given positive parameters.

Under the hypotheses of Theorem  3.1 , there exists \(\Gamma _{0}> 0\) such that for any \(\Gamma> \Gamma _{0}\) the approximate problem ( 3.45 )–( 3.60 ) admits a strong solution {ϱ, u, ϑ} belonging to the following regularity class:

Remark

As a matter of fact, since the velocity field u is continuously differentiable, a bootstrap argument could be used in order to show that ϑ is smooth, hence a classical solution of (3.55) for t > 0, as soon as the thermodynamic functions p, e as well as the transport coefficients μ, λ, and κ are smooth functions of ϱ, ϑ on the set (0, ∞)².

In spite of a considerable number of technicalities, the proof of Proposition 3.1 is based on standard arguments. We adopt the following strategy:

• The solution u of the approximate momentum equation (3.49) is looked for as a fixed point of a suitable integral operator in the Banach space C([0, T]; X _n ). Consequently, the functions ϱ, ϑ have to be determined in terms of u. This is accomplished in the following manner:

• Given u, the approximate continuity equation (3.45) is solved directly by means of the standard theory of linear parabolic equations.

• Having solved (3.45)–(3.47) we determine the temperature ϑ as a solution of the quasilinear parabolic problem (3.55)–(3.57), where ϱ, u play a role of given data.

3.4.1 Approximate Continuity Equation

The rest of this section is devoted to the proof of Proposition 3.1. We start with a series of preparatory steps. Following the strategy delineated in the previous paragraph, we fix a vector field u and discuss solvability of the Neumann-initial value problem (3.45)–(3.47).

■       Approximate Continuity Equation:

Lemma 3.1

Let \(\Omega \subset \mathbb{R}^{3}\) be a bounded domain of class C ^2,ν , ν ∈ (0, 1) and let u ∈ C([0, T]; X _n ) be a given vector field. Suppose that ϱ _0,δ belongs to the class of regularity specified in ( 3.48 ).

Then problem ( 3.45 )–( 3.47 ) possesses a unique classical solution ϱ = ϱ _u , more specifically,

for a certain ν > 0. Moreover, the mapping u ∈ C([0, T]; X _n ) ↦ ϱ _u maps bounded sets in C([0, T]; X _n ) into bounded sets in V and is continuous with values in \(C^{1}([0,T] \times \overline{\Omega })\) .

Finally,

where \(\underline{\varrho }_{0} =\inf _{\Omega }\varrho _{0,\delta }\) , \(\overline{\varrho }_{0} =\sup _{\Omega }\varrho _{0,\delta }.\)

Proof

 Step 1: The unique strong solution of problem (3.45)–(3.48)

that satisfies the estimate

with \(c = c(\varepsilon,T,\|\mathbf{u}\|_{C[0,T];C^{\nu }(\overline{\Omega }))})> 0\), may be constructed by means of the standard Galerkin approximation within the standard L ² theory.

The maximal L ^p − L ^q regularity resumed in Theorem 11.29 in Appendix applied to the problem

combined with a bootstrap argument gives the bound

for any p > 3.

Since \(W^{2-\frac{2} {p},p}(\Omega )\hookrightarrow C^{1,\nu }(\overline{\Omega })\) for any sufficiently large p, we have div _x (ϱ u) ∈ C([0, T]; \(C^{1,\nu }(\overline{\Omega }))\) and may employ Theorem 11.30 from Appendix to show relation (3.62) as well as boundedness of the map u ↦ ϱ _u : C([0, T]; X _n ) → V.

Step 2: The difference \(\omega =\varrho _{\mathbf{u}_{1}} -\varrho _{\mathbf{u}_{2}}\) satisfies

Similar reasoning as in the first step applied to this equation yields the continuity of the map u ↦ ϱ _u from C([0, T]; X _n ) to \(C^{1}([0,T] \times \overline{\Omega })\).

Step 3: The difference

obeys a differential inequality

When multiplied on the positive part | ω |⁺ and integrated over \(\Omega\), the first relation gives \(\|\,\vert \omega \vert ^{+}(t)\|_{L^{2}(\Omega )} \leq 0\) which shows the right inequality in (3.63). The left inequality can be obtained in a similar way. Lemma 3.1 is thus proved. The reader may consult [102, Chap. 7.3] or [224, Sect. 7.2] for more details. □

3.4.2 Approximate Internal Energy Equation

Having fixed u, together with ϱ = ϱ _u —the unique solution of problem (3.45)–(3.47)—we focus on the approximate internal energy equation (3.55) that can be viewed as a quasilinear parabolic problem for the unknown ϑ.

Comparison Principle To begin, we establish a comparison principle in the class of strong (super, sub) solutions of problem (3.55)–(3.57). We recall that a function ϑ is termed a super (sub) solution if it satisfies (3.55) with “ = ” sign replaced by “ ≥ ” (“ ≤ ”).

Lemma 3.2

Given the quantities

assume that \(\underline{\vartheta }\) and \(\overline{\vartheta }\) are respectively a sub and super-solution to problem ( 3.55 )–( 3.57 ) belonging to the class

and satisfying

Then

Proof

As \(\underline{\vartheta }\), \(\overline{\vartheta }\) belong to the regularity class specified in (3.66), we can compute

where we have introduced

and where we have set

In accordance with our hypotheses, we may assume that F = F(t, x, ϑ) is globally Lipschitz with respect to ϑ.

Denoting | z |⁺ = max{z, 0} the positive part, we have

for any \(w \in W^{1,2}((0,T) \times \Omega )\), in particular,

Moreover, as both e _δ and \(\mathcal{K}_{\delta }\) are increasing functions of ϑ, we have

Seeing that

we can integrate (3.69) in order to deduce

for any τ > 0. Here we have used Lipschitz continuity of F(t, x, ⋅ ) and the fact that \(\vert \underline{\vartheta }-\overline{\vartheta }\vert\) \(\mathrm{sgn}^{+}[\varrho e_{\delta }(\varrho,\underline{\vartheta }) -\varrho e_{\delta }(\varrho,\overline{\vartheta })]\) \(\leq c\vert \varrho e_{\delta }(\varrho,\underline{\vartheta }) -\varrho e_{\delta }(\varrho,\overline{\vartheta })\vert ^{+}\) which follows from (3.9), (3.11), (3.65), (3.67). Thus a direct application of Gronwall’s lemma, together with the monotonicity of e _δ with respect to ϑ, completes the proof.

□

Corollary 3.1

For given data ϱ, u satisfying ( 3.65 ), and a measurable function ϑ _0,δ such that

problem ( 3.55 )–( 3.57 ) admits at most one (strong) solution ϑ in the class specified in ( 3.66 )–( 3.67 ).

Another application of Lemma 3.2 gives rise to uniform bounds on the function ϑ in terms of the data.

Corollary 3.2

Let ϱ, u belong to the regularity class ( 3.65 ), and let ϑ _0,δ satisfy ( 3.70 ). Suppose that ϑ is a (strong) solution of problem ( 3.55 )–( 3.57 ) belonging to the regularity class ( 3.66 ).

Then there exist two constants \(\underline{\vartheta }\) , \(\overline{\vartheta }\) depending only on the quantities

satisfying

and

Proof

It is a routine matter to check that a constant function \(\underline{\vartheta }\) is a subsolution of (3.55)–(3.57) as soon as

Revoking (3.30) we can use hypotheses (3.65), (3.13), together with estimate (3.32), in order to see that all quantities on the right-hand side of (3.73) are bounded in terms of \(\|\varrho \|_{ C^{1}([0,T]\times \overline{\Omega })}\) and \(\|\mathbf{u}\|_{C([0,T];X_{n})}\) provided, say, \(0 <\underline{\vartheta }<1\). Note that all norms are equivalent when restricted to the finite dimensional space X _n .

Consequently, a direct application of the comparison principle established in Lemma 3.2 yields the left inequality in (3.72).

Following step by step with obvious modifications the above procedure, the upper bound claimed in (3.72) can be established by help of the dominating term −ɛϑ ⁵ in (3.55).

□

Remark

Corollary 3.2 reveals the role of the extra term δ∕ϑ ² in Eq. (3.55), namely to keep the absolute temperature ϑ bounded below away from zero at this stage of the approximation procedure. Positivity of ϑ is necessary for the passage from (3.55) to the entropy balance equation used in the weak formulation of the Navier-Stokes-Fourier system.

Priori Estimates We shall derive a priori estimates satisfied by any strong solution of problem (3.55)–(3.57).

Lemma 3.3

Let the data ϱ, u belong to the regularity class ( 3.65 ), and let \(\vartheta _{0,\delta } \in W^{1,2}(\Omega )\) satisfy ( 3.70 ).

Then any strong solution ϑ of problem ( 3.55 )–( 3.57 ) belonging to the class ( 3.66 )–( 3.67 ) satisfies the estimate

where h is bounded on bounded sets.

Proof

Note that relation (3.74) represents the standard energy estimates for problem (3.55)–(3.57). These are easily deduced via multiplying Eq. (3.55) by ϑ and integrating the resulting expression by parts in order to obtain

where

In view of the uniform bounds already proved in (3.72), the function F ₁ is bounded in \(L^{\infty }((0,T) \times \Omega )\) in terms of the data.

Similarly, multiplying (3.55) on \(\partial _{t}\mathcal{K}_{\delta }(\vartheta )\) gives rise to

where

is bounded in \(L^{\infty }((0,T) \times \Omega )\) in terms of the data.

Taking the sum of (3.75), (3.76), and using Young’s inequality and Gronwall’s lemma, we conclude that

Finally, evaluating \(\Delta _{x}\mathcal{K}_{\delta }(\vartheta )\) by means of Eq. (3.55), we get (3.74).

□

Existence for the Approximate Internal Energy Equation Having prepared the necessary material, we are ready to show existence of strong solutions to problem (3.55)–(3.57). In fact, the a priori bounds (3.72), (3.74) imply compactness of solutions in the space \(L^{2}(0,T;W^{1,2}(\Omega ))\), in particular, any accumulation point of a family of strong solutions is another solution of the same problem. Under these circumstances, showing existence is a routine matter. Regularizing the data ϱ, u with respect to the time variable, and approximating the quantities μ, η, κ _δ , e, p by smooth ones as the case may be, we can construct a family of approximate solutions to problem (3.55)–(3.57) via the classical results for quasilinear parabolic equations. Then we pass to the limit in a suitable sequence of approximate solutions to recover the (unique) solution of problem (3.55)–(3.57). The relevant theory of quasilinear parabolic equations taken over from the book (Ladyzhenskaya et al. [179, Chap. V]) is summarized in Sect. 11.16 in Appendix.

Hereafter we describe a possible way of the construction of the approximations to problem (3.55)–(3.57).

1. (i)

   Let ν ∈ (0, 1) be the same parameter as in Lemma 3.1. To begin, we extend \(\varrho \in C([0,T];C^{2,\nu }(\overline{\Omega })) \cap C^{1}([0,T];C^{0,\nu }(\overline{\Omega }))\), u ∈ C([0, T]; X _n ), continuously to \(\varrho \in C(\mathbb{R};C^{2,\nu }(\overline{\Omega })) \cap C^{1}(\mathbb{R};C^{0,\nu }(\overline{\Omega }))\), \(\mathrm{supp}\varrho \subset (-2T,2T) \times \overline{\Omega }\), \(\mathbf{u} \in C(\mathbb{R},X_{n})\), \(\mathrm{supp}\mathbf{u} \subset (-2T,2T) \times \overline{\Omega }\). We approximate \(\mathcal{Q}_{\delta }\) by smooth functions \(\mathcal{Q}_{\omega }\) on \([0,T] \times \overline{\Omega }\) and we take sequence of initial conditions

   $$\displaystyle{C^{2,\nu }(\overline{\Omega }) \ni \vartheta _{ 0,\omega } \rightarrow \vartheta _{0,\delta }\;\mbox{ in}W^{1,2}(\Omega ) \cap L^{\infty }(\Omega )}$$

   such that \(\inf _{x\in \Omega }\vartheta _{0,\omega }(x)>\underline{\vartheta } _{0}\) uniformly with respect to ω → 0+, where \(\underline{\vartheta }_{0}\) is a positive constant.

2. (ii)

   We denote

   $$\displaystyle{E_{M}(\varrho,\vartheta ) =\varrho e_{M}(\varrho,\vartheta )}$$

   and set

   $$\displaystyle{ E_{\delta,\omega }(\varrho,\vartheta ) = [<E_{M}>]^{\omega }(\varrho,\theta _{\omega }) + a\theta _{\omega }^{4}+\delta \varrho \vartheta, }$$

   (3.77)
   $$\displaystyle{\{\partial _{\vartheta }E\}_{\delta,\omega }(\varrho,\vartheta ) = [<\partial _{\vartheta }E_{M}>]^{\omega }(\varrho,\vartheta ) + 4a\, \frac{\vartheta ^{4}} {\sqrt{\vartheta ^{2 } +\omega ^{2}}}+\delta \varrho }$$
   $$\displaystyle{\kappa _{\delta,\omega }(\vartheta ) = [<\kappa _{M}>]^{\omega }(\theta _{\omega }) + [<\kappa _{R}>]^{\omega }(\theta _{\omega }) +\delta (\theta _{\omega }^{\Gamma } + \frac{1} {\sqrt{\vartheta ^{2 } +\omega ^{2}}}),}$$
   $$\displaystyle{\mathcal{K}_{\delta,\omega }(\vartheta ) =\int _{ 1}^{\vartheta }\kappa _{ \delta,\omega }(\tau )\,d\tau,}$$
   $$\displaystyle{p_{\omega }(\varrho,\vartheta ) = [<p_{M}>]^{\omega }(\varrho,\theta _{\omega }) + \frac{a} {3}\theta _{\omega }^{4},}$$
   $$\displaystyle{G(t,x) =\Big ((\Gamma \varrho ^{\Gamma -2} + 2)\vert \nabla _{ x}\varrho \vert ^{2}\Big)(t,x),\;G_{\omega }(t,x) = G^{\omega }(t,x)}$$
   $$\displaystyle{\mathbb{S}_{\delta,\omega }(\vartheta,\nabla _{x}\mathbf{u}^{\omega }) =<\mu> ^{\omega }(\theta _{\omega })\Big(\nabla \mathbf{u}^{\omega } + \nabla ^{T}\mathbf{u}^{\omega } -\frac{2} {3}\mathrm{div}\mathbf{u}^{\omega }\mathbb{I}\Big)+ <\eta> ^{\omega }(\theta _{\omega })\mathrm{div}\mathbf{u}^{\omega }\mathbb{I},}$$

   where

   $$\displaystyle{ \begin{array}{c} \theta _{\omega } =\theta _{\omega }(\vartheta ) = \frac{\sqrt{\vartheta ^{2 } +\omega ^{2}}} {1+\omega \sqrt{\vartheta ^{2 } +\omega ^{2}}},\\ \\ <a> (z) = \left \{\begin{array}{c} a(z)\;\mbox{ if }z \in (0,\infty )^{N} \\ \\ \max \{\inf _{z\in (0,\infty )^{N}}a(z)\,,\,0\}\end{array} \right \},\;N = 1,2.\end{array} }$$

   (3.78)

   The operator b ↦ b ^ω, ω > 0 is the standard regularizing operator, see (11.4) in Sect. 11.2, that applies to all independent variables in the case of functions < E _M >, < ∂ _ϑ E _M >, < p >, < μ >, < η >, < κ _M >, and to the variable t in the case of functions ϱ(t, x), u(t, x), G(t, x). Notice that in virtue of hypotheses (3.21)–(3.23) and (3.11)

   $$\displaystyle{ \kappa _{\delta,\omega }(\vartheta ) \geq \underline{\kappa }_{M}> 0,\quad \{\partial _{\vartheta }E\}_{\delta,\omega }(\varrho,\vartheta )>\delta \underline{\varrho }> 0 }$$

   (3.79)

   for all \((\varrho,\vartheta ) \in \mathbb{R}^{2}\), where \(\underline{\varrho }=\inf _{(0,T)\times \Omega }\varrho\).

3. (iii)

   We will find a solution of problem (3.55)–(3.57), as a limit of the sequence {ϑ _ω } _{ω > 0} of solutions to the following equation

   $$\displaystyle{\{\partial _{\vartheta }E\}_{\delta,\omega }(\varrho ^{\omega },\vartheta )\partial _{t}\vartheta + \mathrm{div}\Big(E_{\delta,\omega }(\varrho ^{\omega },\vartheta )\mathbf{u}\Big) - \Delta _{x}\mathcal{K}_{\delta,\omega }(\vartheta )}$$
   $$\displaystyle{ = -\partial _{\varrho }E_{\delta,\omega }(\varrho ^{\omega },\vartheta )\partial _{t}\varrho ^{\omega } + \mathbb{S}_{\delta,\omega }(\nabla _{x}\mathbf{u}^{\omega },\vartheta ): \nabla \mathbf{u}^{\omega }+ }$$

   (3.80)
   $$\displaystyle{\varepsilon \delta G_{\omega } - p_{\omega }(\varrho ^{\omega },\vartheta ) - \frac{\delta } {\vartheta ^{2} +\omega ^{2}} +\varepsilon \theta _{ \omega }^{5} +\varrho ^{\omega }\mathcal{Q}_{\omega },}$$
   $$\displaystyle{\nabla _{x}\vartheta \cdot \mathbf{n}\vert _{\partial \Omega } = 0,\;\vartheta (0,x) =\vartheta _{0,\omega }(x).}$$

   Problem (3.80) for the unknown ϑ has the form of the following quasilinear parabolic equation

   $$\displaystyle{ \begin{array}{c} \partial _{t}\vartheta -\sum _{i,j=1}^{3}a_{ij}(t,x,\vartheta )\partial _{x_{i}}\partial _{x_{j}}\vartheta + b(t,x,\vartheta,\nabla _{x}\vartheta ) = 0\quad \mbox{ in }(0,T) \times \Omega, \\ \Big(\sum _{i,j=1}^{3}a_{ij}\partial _{x_{j}}\vartheta \,n_{i}+\psi \Big)\Big\vert _{(0,T)\times \partial \Omega } = 0, \\ \vartheta \vert _{\{0\}\times \Omega } = 0,\end{array} }$$

   (3.81)

   where

   $$\displaystyle{ a_{ij}(t,x,\vartheta ) = \frac{\kappa _{\delta,\omega }(\vartheta )} {[\partial _{\vartheta }E]_{\delta,\omega }(\varrho ^{\omega }(t,x),\vartheta )}\delta _{ij},\quad i,j = 1,2,3,\quad \psi = 0 }$$

   (3.82)

   and

   $$\displaystyle{ b(t,x,\vartheta,\mathbf{z}) = \frac{1} {\{\partial _{\vartheta }E\}_{\delta,\omega }(\varrho ^{\omega }(t,x),\vartheta )}\Big[-\kappa _{\delta,\omega }^{{\prime}}(\vartheta )\vert \mathbf{z}\vert ^{2}+ }$$

   (3.83)
   $$\displaystyle{\partial _{\varrho }E_{\delta,\omega }(\varrho ^{\omega }(t,x),\vartheta )\partial _{t}\varrho ^{\omega }(t,x) + \partial _{\varrho }E_{\delta,\omega }(\varrho ^{\omega }(t,x),\vartheta )\big(\nabla \varrho ^{\omega }\cdot \mathbf{u}^{\omega }\big)(t,x)+}$$
   $$\displaystyle{-\mathbb{S}_{\delta,\omega }(\nabla _{x}\mathbf{u}^{\omega }(t,x),\vartheta ): \nabla \mathbf{u}^{\omega }(t,x) + \partial _{\vartheta }E_{\delta,\omega }(\varrho ^{\omega }(t,x),\vartheta )\big(\mathbf{z} \cdot \mathbf{u}^{\omega }\big)(t,x)+}$$
   $$\displaystyle{E_{\delta,\omega }(\varrho ^{\omega }(t,x),\vartheta )\mathrm{div}_{x}\mathbf{u}^{\omega } + p_{\omega }(\varrho ^{\omega }(t,x),\vartheta )\mathrm{div}_{x}\mathbf{u}^{\omega }(t,x)}$$
   $$\displaystyle{-\epsilon \delta G_{\omega }(t,x) + \frac{\delta } {\vartheta ^{2} +\omega ^{2}} -\varepsilon \theta _{\omega }^{5}(\vartheta ) -\varrho ^{\omega }\mathcal{Q}_{\omega }(t,x)].}$$

   In accordance with the properties of mollifiers recalled in Sect. 11.2 in Appendix, a _ij , b, ψ satisfy assumptions of Theorem 11.31 from Sect. 11.16. Therefore, problem (3.80) admits a (unique) solution ϑ = ϑ _ω which belongs to class

   $$\displaystyle{\vartheta _{\omega } \in C([0,T];C^{2,\nu }(\overline{\Omega })) \cap C^{1}([0,T] \times \overline{\Omega }),\quad \partial _{ t}\vartheta _{\omega } \in C^{0,\nu /2}([0,T];C(\overline{\Omega })).}$$
4. (iv)

   The proofs of Lemma 3.2, Corollary 3.2 and Lemma 3.3 apply with minor modifications to system (3.80), yielding the uniform bounds

   $$\displaystyle{\|\frac{1} {\vartheta _{\omega }} \|_{L^{\infty }((0,T)\times \Omega )} +\|\vartheta _{\omega }\|_{L^{\infty }((0,T)\times \Omega )} \leq c,}$$
   $$\displaystyle{\mathrm{ess}\sup _{t\in (0,T)}\|\vartheta _{\omega }\|_{W^{1,2}(\Omega )}^{2} +\int _{ 0}^{T}\Big(\|\partial _{ t}\vartheta _{\omega }\|_{L^{2}(\Omega )}^{2} +\| \Delta _{ x}\mathcal{K}_{\delta }(\vartheta _{\omega })\|_{L^{2}(\Omega )}^{2}\Big)\mathrm{d}t \leq c}$$

   with respect to ω → 0+. With these bounds and the properties of mollifiers recalled in Sect. 11.2 at hand, the limit passage from system (3.80) to (3.55)–(3.57) is an easy exercise.

The results achieved in this section can be stated as follows.

■      Approximate Internal Energy Equation:

Lemma 3.4

Let \(\Omega \subset \mathbb{R}^{3}\) be a bounded domain of class C ^2,ν , ν ∈ (0, 1). Let u ∈ C([0, T]; X _n ) be a given vector field and let ϱ = ϱ _u be the unique solution of the approximate problem ( 3.45 )–( 3.47 ) constructed in Lemma  3.1 . Further

1. (i)

   let the initial datum \(\vartheta _{0,\delta } \in W^{1,2}(\Omega ) \cap L^{\infty }(\Omega )\) be bounded below away from zero as stated in hypothesis ( 3.58 ) and the source term \(\mathcal{Q}_{\delta }\) satisfies ( 3.60 );

2. (ii)

   let the constitutive functions p, e, s and the transport coefficients μ, η, κ obey the structural assumptions ( 3.7 )–( 3.23 ).

Then problem ( 3.55 )–( 3.57 ), with e _δ , \(\mathcal{K}_{\delta }\) defined in ( 3.59 ) and u , ϱ _u fixed, possesses a unique strong solution ϑ = ϑ _u belonging to the regularity class

Moreover, the mapping u → ϑ _u maps bounded sets in C([0, T]; X _n ) into bounded sets in Y and is continuous with values in \(L^{2}(0,T;W^{1,2}(\Omega ))\) .

3.4.3 Local Solvability of the Approximate Problem

At this stage, we are ready to show the existence of approximate solutions on a possibly short time interval (0, T _max). In accordance with (3.50), X _n is a finite dimensional subspace of \(L^{2}(\Omega, \mathbb{R}^{3})\) endowed with the Hilbert structure induced by \(L^{2}(\Omega; \mathbb{R}^{3})\). We denote by P _n the orthogonal projection of \(L^{2}(\Omega, \mathbb{R}^{3})\) onto X _n . Furthermore, we set

We start rewriting (3.49) as a fixed point problem:

where we have denoted

and

Note that

and

where c > 0 depends solely on n, in particular, it is independent of the data specified in (2.41) and the parameters ɛ, δ, \(\Gamma\).

Given u ∈ C([0, T]; X _n ), the density ϱ = ϱ _u can be identified as the unique (classical) solution of the parabolic problem (3.45)–(3.48), the existence of which is guaranteed by Lemma 3.1. In particular, the (approximate) density ϱ _u remains bounded below away from zero as soon as we can control div _x u. Note that, at this level of approximation, the norm of div _x u is dominated by that of u as the dimension of X _n is finite.

With u, ϱ _u at hand, the temperature ϑ = ϑ _u can be determined as the unique solution of problem (3.55)–(3.57) constructed by means of Lemma 3.4, in particular, ϑ is strictly positive with a lower bound in terms of the data, see Corollary 3.2.

If \(\|\mathbf{u}\|_{C([0,T];X_{n})} \leq R\), then

where we have used Lemmas 3.1, 3.4, specifically, bounds (3.62), (3.84). The constant c ₀, determined in terms of equivalence of norms on X _n , depends solely on n and h is a positive function bounded on bounded sets.

Consequently, if

the operator u ↦ S[u] determined through (3.86) maps the ball

into itself as soon as τ ₀ is small enough.

Moreover, as a consequence of (3.88) and smoothness of ϱ, the image of \(B_{R,\tau _{0}}\) consists of uniformly Lipschitz functions on [0, τ ₀], in particular, it belongs to a compact set in C([0, τ ₀]; X _n ). Thus a direct application of the Leray-Schauder fixed point theorem yields existence of a solution {ϱ, u, ϑ} of the approximate problem (3.45)–(3.57) defined on a (possibly short) time interval [0, T(n)]. Finally, taking advantage of Lemma 3.1, we deduce from (3.86) that

The above procedure can be iterated as many times as necessary to reach T(n) = T as long as there is a bound on u independent of T(n). The existence of such a bound is the main topic discussed in the next section.

3.4.4 Uniform Estimates and Global Existence

Let {ϱ, u, ϑ} be an approximate solution of problem (3.45)–(3.57) defined on a time interval [0, T _max], T _max ≤ T. The last step in the proof of Proposition 3.1 is to establish a uniform (in time) bound on the norm \(\|\mathbf{u}(t)\|_{X_{n}}\) for t ∈ [0, T _max] independent of T _max. The existence of such a bound allows us to iterate the local construction described in the previous section in order to obtain an approximate solution defined on the full time interval [0, T]. To this end, the a priori estimates derived in Sect. 2.2 will be adapted in order to accommodate the extra terms arising at the actual level of approximation.

First of all, it follows from (3.45), (3.46) that the total mass remains constant in time, specifically,

The next observation is that the quantity ψ u, with ψ = ψ(t), ψ ∈ C _c ¹[0, T _max), can be taken as a test function in the variational formulation of the momentum equation (3.49) to obtain

which, combined with (3.55), gives rise to the approximate energy balance

Moreover, dividing the approximate internal energy equation (3.55) on ϑ, we obtain, after a straightforward manipulation, an approximate entropy production equation in the form

satisfied a.a. in \((0,T_{\mathrm{max}}) \times \Omega\), where

and

Relations (3.95), (3.96) give rise to uniform estimates similar to those obtained in Sect. 2.2.3. Indeed, multiplying (3.96) on \(\overline{\vartheta }\), where \(\overline{\vartheta }\) is a arbitrary positive constant, integrating over \(\Omega\), and subtracting the resulting expression from (3.96), we get

where \(H_{\delta,\overline{\vartheta }}\) is an analogue of the Helmholtz function introduced in (2.48), specifically,

Here, in accordance with (3.98),

where, by virtue of Gibbs’ relation (3.7),

Equality (3.99) therefore transforms to

where

Similarly to Sect. 2.2.3, relation (3.104) provides all the necessary uniform estimates as soon as we check that the terms on the right hand side can be controlled by the positive quantities on the left hand side. In order to see that, observe that the term δ∕ϑ ² on the right-hand side of (3.104) is dominated by its counterpart δ∕ϑ ³ in the entropy production term σ _ɛ, δ . Analogously, the quantity \(\varepsilon \overline{\vartheta }\vartheta ^{4}\) at the right hand side is “absorbed” by the term ɛϑ ⁵ at the left hand side of (3.104). Finally, the term \(\varrho (1 -\frac{\overline{\vartheta }}{\vartheta })Q_{\delta }\) can be written as a sum \(\varrho (1 -\frac{\overline{\vartheta }}{\vartheta })Q_{\delta }1_{\{\vartheta \leq 1\}} +\varrho (1 -\frac{\overline{\vartheta }}{\vartheta })Q_{\delta }1_{\{\vartheta>1\}}\), where \(\int _{0}^{\tau }\int _{\Omega }\varrho (1 -\frac{\overline{\vartheta }}{\vartheta })Q_{\delta }1_{\{\vartheta \leq 1\}}\,\mathrm{d}x\,\mathrm{d}t \leq 0\), while \(\vert \int _{0}^{\tau }\int _{\Omega }\varrho (1 -\frac{\overline{\vartheta }}{\vartheta })Q_{\delta }1_{\{\vartheta>1\}}\,\mathrm{d}x\,\mathrm{d}t\vert\) is bounded by \(\overline{\varrho }T\vert \Omega \vert \|Q_{\delta }\|_{L^{\infty }((0,T)\times \Omega )}\).

Consequently, it remains to handle the quantity

appearing on the right-hand side of (3.104). To this end, we first use hypothesis (3.13), together with (3.30), in order to obtain

where, furthermore,

and, similarly,

Thus we infer that

provided ɛ = ɛ(δ) > 0 is small enough.

Taking into account the properties of the function \(H_{\delta,\overline{\vartheta }}\) (see (2.49)–(2.50) in Sect. 2.2.3), we are ready to summarize the so far obtained estimates as follows:

where c is a positive constant depending on the data specified in (2.41) but independent of T _max, n, ɛ, and δ.

At this stage, following the line of arguments presented in Sect. 2.2.3, we can use the bounds listed in (3.107) in order to deduce uniform estimates on the approximate solutions defined on the time interval [0, T _max] independent of T _max. Indeed it follows from (3.107) that

in particular, by means of hypothesis (3.53) and Proposition 2.1,

Consequently, by virtue of (3.63), the density ϱ is bounded below away from zero uniformly on [0, T _max], and we conclude

As already pointed out, bound (3.109) and the local construction described in the previous section give rise to an approximate solution {ϱ, u, ϑ} defined on [0, T]. We have proved Proposition 3.1.

In the previous section, we constructed a family of approximate solutions to the Navier-Stokes-Fourier system satisfying (3.45)–(3.60), see Proposition 3.1. Our goal in the remaining part of this chapter is to examine successively the asymptotic limit for n → ∞, ɛ → 0, and, finally, δ → 0. The first step of this rather long procedure consists in performing the limit n → ∞.

We recall that the spaces X _n introduced in Sect. 3.3.1 are formed by sufficiently smooth functions \(\boldsymbol{\varphi }\) (belonging at least to \(C^{2,\nu }(\overline{\Omega })\)) satisfying either the complete slip boundary condition (3.51) or the no-slip boundary conditions (3.52) as the case may be. Clearly, the approximate velocity field u ∈ C ¹([0, T]; X _n ) belongs to the same class for each fixed t ∈ [0, T]. In the remaining part of the chapter, we make an extra hypothesis that the vector space X,

for any 1 ≤ p < ∞, where

Such a choice of X _n is possible provided \(\Omega\) belongs to the regularity class C ^2,ν required by Theorem 3.1. The interested reader may consult Sect. 11.8 in Appendix for technical details.

3.5.1 Estimates Independent of the Dimension of Faedo-Galerkin Approximations

For ɛ > 0, δ > 0 fixed, let {ϱ _n , u _n , ϑ _n } _{n = 1} ^∞ be a sequence of approximate solutions constructed in Sect. 3.4. In accordance with (3.107), this sequence admits the following uniform estimates:

and

where c denotes a generic constant depending only on the data specified in (2.41), in particular, c is independent of the parameters n, ɛ, and δ.

By virtue of the coercivity properties of \(H_{\delta,\overline{\vartheta }}\) established in (2.49), (2.50), the uniform bound (3.110) implies that

therefore we can assume

On the other hand, estimate (3.111), together with hypothesis (3.53) and Proposition 2.1, yield

in particular

at least for a suitable subsequence.

At this point it is worth noting that the limit density ϱ is still a non-negative quantity albeit not necessarily strictly positive as this important property stated in (3.63) is definitely lost in the limit passage due to the lack of suitable uniform estimates for div _x u _n . The fact that the class of weak solutions admits cavities (vacuum regions) seems rather embarrassing from the point of view of the model derived for non-dilute fluids, but still physically acceptable.

Convergence (3.115) can be improved to

as ϱ _n , u _n solve Eq. (3.45). Indeed we check easily that for all \(\varphi \in C_{c}^{\infty }(\Omega )\), the functions \(t \rightarrow [\int _{\Omega }\varrho _{n}\varphi \,\mathrm{d}x](t)\) form a bounded and equi-continuous sequence in C[0, T]. Consequently, the standard Arzelà-Ascoli theorem (Theorem 1) yields

Since ϱ _n satisfy (3.114), the convergence extends easily to each \(\varphi \in L^{\Gamma '}(\Omega )\) via density.

In order to deduce uniform estimates on the approximate temperature ϑ _n , we exploit the structural properties of the Helmholtz function \(H_{\overline{\vartheta }}\). Note that these follow directly from the hypothesis of thermodynamics stability and as such may be viewed as a direct consequence of natural physical principles. The following assertion will be amply used in future considerations.

■     Coercivity of the Helmholtz Function:

Proposition 3.2

Let the functions p, e, and s be interrelated through Gibbs’ equation ( 1.2 ), where p and e comply with hypothesis of thermodynamic stability (1.44).

Then for any fixed \(\overline{\varrho }> 0\) , \(\overline{\vartheta }> 0\) , the Helmholtz function

satisfies

for all positive ϱ, ϑ.

Proof

As the result obviously holds if s(ϱ, ϑ) ≤ 0, we focus on the case s(ϱ, ϑ) > 0. It follows from (2.49), (2.50) that

whence

and, similarly,

Summing up the last two inequalities we obtain the desired conclusion.

□

On the basis of Proposition 3.2, we can deduce from hypothesis (3.9) and the total energy estimate (3.110) that

therefore we may assume

In addition, using boundedness of the entropy production rate stated in (3.111) we get

Estimates (3.119), (3.121), together with Poincare’s inequality formulated in terms of Proposition 2.2, yield

in particular,

Moreover, by virtue of estimate (3.111), we have

notably the limit function ϑ is positive almost everywhere in \((0,T) \times \Omega\) and satisfies

where we have used convexity of the function z ↦ z ⁻³ on (0, ∞), see Theorem 11.27 in Appendix.

Finally, the standard embedding relation \(W^{1,2}(\Omega )\hookrightarrow L^{6}(\Omega )\), together with (3.121), can be used in order to derive higher integrability estimates of ϑ _n , namely

Note that, as a byproduct of (3.125), (3.126),

3.5.2 Limit Passage in the Approximate Continuity Equation

At this stage, we are ready to show strong (pointwise) convergence of the approximate densities and to let n → ∞ in equation (3.45). To this end, we need to control the term pdiv _x u in the approximate energy balance (3.94).

A direct application of (3.32) yields

where, by virtue of (3.114), (3.116), (3.119), and (3.126), the last integral is bounded provided \(\Gamma> 5\). Accordingly, relation (3.94) gives rise to

with c independent of n. Applying the Poincaré inequality (see Proposition 2.2) we get

and

The next step is to obtain uniform estimates on ∂ _t ϱ _n , \(\Delta \varrho _{n}\). This is a delicate task as

where, in accordance with (3.116), (3.129), ∇ _x ϱ _n ⋅ u _n is bounded in \(L^{1}(0,T;L^{\frac{3} {2} }(\Omega ))\), notably this quantity is merely integrable with respect to time. To overcome this difficulty, multiply Eq. (3.45) on G′(ϱ _n ) and integrate by parts to obtain

This is of course nothing other than an integrated “parabolic” version of the renormalized continuity equation (2.2). Taking G(ϱ _n ) = ϱ _n log(ϱ _n ) we easily deduce

As a consequence of (3.110), the kinetic energy is bounded, specifically,

whence estimate (3.132) can be used to obtain

where the product on the right-hand side is bounded in L ²(0, T). Then a standard interpolation argument implies

Applying the L ^p − L ^q theory to the parabolic equation (3.45) (see Sect. 11.15 in Appendix) we conclude that

Now we are ready to carry out the limit passage n → ∞ in the approximate continuity Eq. (3.45). To begin, the uniform bounds established (3.135), together with the standard compactness embedding relations for Sobolev spaces, imply

Moreover, in view of (3.99), (3.117), (3.134), (3.135), and (3.136), it is easy to let n → ∞ in the approximate continuity Eq. (3.45) to obtain

where ϱ is a non-negative function satisfying

together with the initial condition

where ϱ _0,δ has been specified in (3.48).

Our next goal is to show strong convergence of the gradients ∇ _x ϱ _n . To this end, we use the “renormalized” identity (3.131) with G(z) = z ², together with the pointwise convergence established in (3.136), to deduce

for any 0 < τ ≤ T. On the other hand, multiplying Eq. (3.137) on ϱ and integrating by parts yields

whence

3.5.3 Strong Convergence of the Approximate Temperatures and the Limit in the Entropy Equation

Strong Convergence of the Approximate Temperatures The next step is to perform the limit in the approximate entropy balance (3.96). Here the main problem is to show strong (pointwise) convergence of the temperature. Indeed all estimates on {ϑ _n } _{n = 1} ^∞ established above concern only the spatial derivatives leaving open the question of possible time oscillations. Probably the most elegant way to overcome this difficulty is based on the celebrated Div-Curl lemma discovered by Tartar [254].

■     Div-Curl Lemma:

Proposition 3.3

Let \(Q \subset \mathbb{R}^{N}\) be an open set. Assume

where

In addition, let

for a certain s > 1.

Then

Proposition 3.3 is proved in Sect. 11.14 in Appendix for reader’s convenience.

□

The basic idea is to apply Proposition 3.3 to the pair of functions

defined on the set \(Q = (0,T) \times \Omega \subset \mathbb{R}^{4}\), where the term r _n ⁽¹⁾, together with the necessary piece of information concerning div _t, x U _n , are provided by Eq. (3.96).

To see this, we observe first that the only problematic term on the right-hand side of (3.96) can be handled as

(cf. (3.101)–(3.103)). Indeed, in accordance with the uniform estimates (3.106), (3.111), the approximate entropy balance equation (3.96) can be now written in the form

where

and

Hence, by virtue of the uniform estimates (3.106), (3.111)–(3.113), and (3.119),

is bounded in \(L^{1}((0,T) \times \Omega )\), therefore precompact in \(W^{-1,s}((0,T) \times \Omega )\) provided \(s \in [1, \frac{4} {3})\) (cf. Sect. 7). On the other hand, due to (3.116), curl _t, x V _n is obviously bounded in \(L^{2}((0,T) \times \Omega; \mathbb{R}^{4})\) which is compactly embedded into \(W^{-1,2}((0,T) \times \Omega; \mathbb{R}^{4})\). Let us remark that the “space-time” operator curl _t, x applied to the vector field [ϑ _n , 0, 0, 0] involves only the partial derivatives in the spatial variable x.

Consequently, in order to apply Proposition 3.3 in the situation described in (3.141), we have to show that ϱ _n s(ϱ _n , ϑ _n ) and r _n ⁽¹⁾ are bounded in a Lebesgue space “better” than only L ¹.

To this end, write

where ϱ _n s _M (ϱ _n , ϑ _n ) satisfies (3.39), therefore

Consequently, thanks to estimates (3.127), (3.129),

Next we observe that (3.111) implies in the way explained in (2.58) that

Furthermore, it follows from (3.111) that

Moreover, estimates (3.124), (3.126) and (3.119) combined with a simple interpolation yield

on condition that \(\Gamma> 6\). From the last two estimates, we deduce that

Finally, the ɛ-dependent quantity contained in r _n ⁽¹⁾ can be handled in the following way:

• Similarly to the proof of formula (3.144), we conclude, by help of estimates (3.126), (3.127), (3.132), that

  $$\displaystyle{ \{s_{\delta }(\varrho _{n},\vartheta _{n})\nabla \varrho _{n}\}_{n=1}^{\infty }\ \mbox{ is bounded in }L^{ \frac{2\Gamma } {\Gamma +6} }((0,T) \times \Omega ) }$$

  (3.146)

  provided \(\Gamma> 6\).

• Since the specific internal energy e _M satisfies (3.30), we have

  $$\displaystyle{\Big\vert \frac{e_{M}(\varrho _{n},\vartheta _{n})} {\vartheta _{n}} \nabla _{x}\varrho _{n}\Big\vert \leq c(1 + \frac{\varrho _{n}^{\frac{2} {3} }} {\vartheta _{n}} )\vert \nabla _{x}\varrho _{n}\vert;}$$

  whence, in accordance with estimates (3.114), (3.124), and (3.129),

  $$\displaystyle{ \Big\{\frac{e_{M}(\varrho _{n},\vartheta _{n})} {\vartheta _{n}} \nabla _{x}\varrho _{n}\Big\}_{n=1}^{\infty }\ \mbox{ is bounded in}\ L^{ \frac{6\Gamma } {5\Gamma +4} }((0,T) \times \Omega; \mathbb{R}^{3}). }$$

  (3.147)

• By virtue of (3.31) and (3.32),

  $$\displaystyle{ \Big\vert \frac{p_{M}(\varrho _{n},\vartheta _{n})} {\varrho _{n}\vartheta _{n}} \nabla _{x}\varrho _{n}\Big\vert \leq c\vert \nabla _{x}\varrho _{n}\vert \Big(1 + \frac{\varrho _{n}^{\frac{2} {3} }} {\vartheta _{n}} \Big), }$$

  (3.148)

  where the right hand side can be controlled exactly as in (3.147).

Having verified the hypotheses of Proposition 3.3 for the vector fields U _n , V _n specified in (3.141), we are allowed to conclude that

provided \(\Gamma> 6\). In formula (3.149) and hereafter, the symbol \(\overline{F(\mathbf{U})}\) denotes a weak L ¹-limit of the sequence of composed functions {F(U _n )} _{n = 1} ^∞ (cf. Sect. 8).

Since the entropy is an increasing function of the absolute temperature, relation (3.149) can be used to deduce strong (pointwise) convergence of the sequence {ϑ _n } _{n = 1} ^∞.

To begin, we recall (3.97), namely

As all three components of the entropy are increasing in ϑ, we observe that

Indeed, as {ϱ _n } _{n = 1} ^∞ converges strongly (see (3.136)) we have

where, as a direct consequence of monotonicity of s _M in ϑ,

see Theorem 11.26 in Appendix. Here, we have used (3.123), (3.136) yielding

The remaining two inequalities in (3.150) can be shown in a similar way.

Combining (3.149), (3.150) we infer that

in particular, at least for a suitable subsequence, we have

(cf. Theorems 11.26, 11.27 in Appendix).

Limit in the Approximate Entropy Equation Our ultimate goal in this section is to let n → ∞ in the approximate entropy Eq. (3.143).

First of all, we estimate the term

in the same way as in (3.106) transforming (3.143) to inequality

As a consequence of (3.136), (3.144), (3.151),

and, in accordance with (3.116),

Since the sequence {ϑ _n } _{n = 1} ^∞ converges a.a. in \((0,T) \times \Omega\), we can use hypotheses (3.21), (3.22), together with estimates (3.119), (3.122), (3.124), (3.126), to get

yielding, in combination with (3.123),

On the other hand, by virtue of relations (3.121), (3.124), (3.126),

where, according to (3.151),

In order to control the ɛ-term on the left hand side of (3.152), we first observe that

where we have used (3.31), (3.32), (3.39).

As a next step, we apply relations (3.122), (3.129), and (3.132), together with the arguments leading to (3.147), in order to deduce boundedness of the quantity

In particular, by virtue of (3.136), (3.140), (3.151), we obtain

Finally, we identify the asymptotic limit for n → ∞ of the approximate entropy production rate represented through the quantities on the right-hand side of (3.152). In accordance with (3.111), we have

bounded in \(L^{2}((0,T) \times \Omega; \mathbb{R}^{3\times 3})\), and in \(L^{2}((0,T) \times \Omega )\), respectively. In particular,

where we have used (3.117) and (3.151).

Similarly,

and, by virtue of (3.111), (3.123) and (3.151),

By the same token, due to (3.112), (3.136), (3.140),

while, by virtue of (3.113), (3.136), (3.140), (3.151),

Finally, as a consequence of (3.136), (3.151), and the bounds established in (3.124), (3.126), (3.130), we have

The convergence results just established are sufficient in order to perform the weak limit for n → ∞ in the approximate entropy balance (3.152). Although we are not able to show strong convergence of the gradients of ϱ, ϑ, and u, the inequality sign in (3.152) is preserved under the weak limit because of lower semi-continuity of convex superposition operators (cf. Theorem 11.27 in Appendix). Consequently, we are allowed to conclude that

where we have set

and

3.5.4 Limit in the Approximate Momentum Equation

With regard to formulas (3.32), (3.53), estimates (3.114), (3.116), (3.119), (3.126), (3.129), (3.130), and the asymptotic limits established in (3.117), (3.136), (3.140), (3.151), it is easy to identify the limit for n → ∞ in all quantities appearing in the approximate momentum equation (3.49) for a fixed test function φ, with the exception of the convective term. Note that, even at this level of approximations, we have already lost compactness of the velocity field in the time variable because of the hypothetical presence of vacuum zones.

To begin, observe that

where we have used the uniform bounds (3.110), (3.116). Thus we have to show

To this end, observe first that

as a direct consequence of estimates (3.114), (3.133), and strong convergence of the density established in (3.136).

Moreover, it can be deduced from the approximate momentum equation (3.49) that the functions

for any fixed ϕ ∈ ∪ _{n = 1} ^∞ X _n . Since the set ∪ _{n = 1} ^∞ X _n is dense in \(L^{5}(\Omega; \mathbb{R}^{3})\) we obtain, by means of the Arzelà-Ascoli theorem, that

On the other hand, as the Lebesgue space \(L^{5/4}(\Omega )\) is compactly embedded into the dual \(W^{-1,2}(\Omega )\), we infer that

Relation (3.170), together with the weak convergence of the velocities in the space \(L^{2}(0,T;W^{1,2}(\Omega; \mathbb{R}^{3}))\) established in (3.117), give rise to (3.168).

3.5.5 The Limit System Resulting from the Faedo-Galerkin Approximation

Having completed the necessary preliminary steps, in particular, the strong convergence of the density in (3.140), and the strong convergence of the temperature in (3.151), we can let n → ∞ in the approximate system (3.45)–(3.60) to deduce that the limit quantities {ϱ, u, ϑ} satisfy:

1. (i)

   Approximate continuity equation:

   $$\displaystyle{ \partial _{t}\varrho + \mathrm{div}_{x}(\varrho \mathbf{u}) = \varepsilon \Delta \varrho \ \mbox{ a.a. in}\ (0,T) \times \Omega, }$$

   (3.171)

   together with the homogeneous Neumann boundary condition

   $$\displaystyle{ \nabla _{x}\varrho \cdot \mathbf{n}\vert _{\partial \Omega } = 0, }$$

   (3.172)

   and the initial condition

   $$\displaystyle{ \varrho (0,\cdot ) =\varrho _{0,\delta }. }$$

   (3.173)
2. (ii)

   Approximate balance of momentum:

   $$\displaystyle{ \int _{0}^{T}\int _{ \Omega }\Big(\varrho \mathbf{u} \cdot \partial _{t}\boldsymbol{\varphi } +\varrho [\mathbf{u} \otimes \mathbf{u}]: \nabla _{x}\boldsymbol{\varphi } +\Big (\,p +\delta (\varrho ^{\Gamma } +\varrho ^{2})\Big)\mathrm{div}_{ x}\boldsymbol{\varphi }\Big)\ \mathrm{d}x\ \mathrm{d}t }$$

   (3.174)
   $$\displaystyle{=\int _{ 0}^{T}\int _{ \Omega }\Big(\varepsilon (\nabla _{x}\varrho \nabla _{x}\mathbf{u}) \cdot \boldsymbol{\varphi } +\mathbb{S}_{\delta }: \nabla _{x}\boldsymbol{\varphi } -\varrho \mathbf{f}_{\delta }\cdot \boldsymbol{\varphi }\Big)\ \mathrm{d}x\ \mathrm{d}t -\int _{\Omega }(\varrho \mathbf{u})_{0}\cdot \boldsymbol{\varphi }\ \mathrm{d}x,}$$

   satisfied for any test function \(\boldsymbol{\varphi }\in C_{c}^{\infty }([0,T) \times \overline{\Omega }; \mathbb{R}^{3})\), where either

   $$\displaystyle{ \boldsymbol{\varphi }\cdot \mathbf{n}\vert _{\partial \Omega } = 0\ \mbox{ in the case of the complete slip boundary conditions,} }$$

   (3.175)

   or

   $$\displaystyle{ \boldsymbol{\varphi }\vert _{\partial \Omega } = 0\ \mbox{ in the case of the no-slip boundary conditions,} }$$

   (3.176)

   and where we have set

   $$\displaystyle{ \mathbb{S}_{\delta } = \mathbb{S}_{\delta }(\vartheta,\nabla _{x}\mathbf{u}) = (\mu (\vartheta )+\delta \vartheta )\Big(\nabla _{x}\mathbf{u} + \nabla _{x}^{\perp }\mathbf{u} -\frac{2} {3}\mathrm{div}_{x}\mathbf{u}\ \mathbb{I}\Big) +\eta (\vartheta )\mathrm{div}_{x}\mathbf{u}\ \mathbb{I}. }$$

   (3.177)
3. (iii)

   Approximate entropy inequality:

   $$\displaystyle{ \int _{0}^{T}\int _{ \Omega }\varrho s_{\delta }(\varrho,\vartheta )\Big(\partial _{t}\varphi + \mathbf{u} \cdot \nabla _{x}\varphi \Big)\ \mathrm{d}x\ \mathrm{d}t +\int _{ 0}^{T}\int _{ \Omega }\Big(\frac{\kappa _{\delta }(\vartheta )\nabla _{x}\vartheta } {\vartheta } + \varepsilon \mathbf{r}\Big) \cdot \nabla _{x}\varphi \ \mathrm{d}x\ \mathrm{d}t }$$

   (3.178)
   $$\displaystyle{+\int _{0}^{T}\int _{ \Omega }\sigma _{\varepsilon,\delta }\varphi \ \mathrm{d}x\ \mathrm{d}t \leq -\int _{\Omega }(\varrho s)_{0,\delta }\varphi (0,\cdot )\ \mathrm{d}x +\int _{ 0}^{T}\int _{ \Omega }\Big(\varepsilon \vartheta ^{4} -\frac{\varrho } {\vartheta }\mathcal{Q}_{\delta }\Big)\varphi \ \mathrm{d}x\ \mathrm{d}t}$$

   for any test function \(\varphi \in C_{c}^{\infty }([0,T) \times \overline{\Omega })\), φ ≥ 0, where we have set

   $$\displaystyle{ s_{\delta }(\varrho,\vartheta ) = s(\varrho,\vartheta )+\delta \log \vartheta,\ \kappa _{\delta }(\vartheta ) =\kappa (\vartheta ) +\delta \Big (\vartheta ^{\Gamma } + \frac{1} {\vartheta } \Big), }$$

   (3.179)

   and

   $$\displaystyle{ \sigma _{\varepsilon,\delta } = \frac{1} {\vartheta } \Big[\mathbb{S}_{\delta }: \nabla _{x}\mathbf{u} +\Big (\frac{\kappa (\vartheta )} {\vartheta } + \frac{\delta } {2}(\vartheta ^{\Gamma -1} + \frac{1} {\vartheta ^{2}} )\Big)\vert \nabla _{x}\vartheta \vert ^{2} +\delta \frac{1} {\vartheta ^{2}} \Big]+ }$$

   (3.180)
   $$\displaystyle{+ \frac{\varepsilon \delta } {2\vartheta }(\Gamma \varrho ^{\Gamma -2} + 2)\vert \nabla _{ x}\varrho \vert ^{2} + \varepsilon \frac{\partial p_{M}} {\partial \varrho } (\varrho,\vartheta )\frac{\vert \nabla _{x}\varrho \vert ^{2}} {\varrho \vartheta },}$$
   $$\displaystyle{\mathbf{r} = -\Big(\vartheta s_{M,\delta }(\varrho,\vartheta ) - e_{M,\delta }(\varrho,\vartheta ) -\frac{p_{M}(\varrho,\vartheta )} {\varrho } \Big)\frac{\nabla _{x}\varrho } {\vartheta }.}$$
4. (iv)

   Approximate total energy balance:

   $$\displaystyle{ \int _{\Omega }\Big(\frac{1} {2}\varrho \vert \mathbf{u}\vert ^{2} +\varrho e_{\delta }(\varrho,\vartheta ) + \delta ( \frac{\varrho ^{\Gamma }} {\Gamma - 1} +\varrho ^{2})\Big)(\tau )\ \mathrm{d}x }$$

   (3.181)
   $$\displaystyle{= \int _{\Omega }\Big(\frac{1} {2} \frac{\vert (\varrho \mathbf{u})_{0,\delta }\vert ^{2}} {\varrho _{0,\delta }} +\varrho _{0,\delta }e_{0,\delta } + \delta ( \frac{\varrho _{0,\delta }^{\Gamma }} {\Gamma - 1} +\varrho _{ 0,\delta }^{2})\Big)\ \mathrm{d}x}$$
   $$\displaystyle{+\int _{0}^{\tau }\int _{ \Omega }\Big(\varrho \mathbf{f}_{\delta } \cdot \mathbf{u} +\varrho \mathcal{Q}_{\delta } +\delta \frac{1} {\vartheta ^{2}} -\varepsilon \vartheta ^{5}\Big)\ \mathrm{d}x\ \mathrm{d}t\ \mbox{ for a.a.}\ \tau \in [0,T],}$$

   where

   $$\displaystyle{ e_{\delta }(\varrho,\vartheta ) = e(\varrho,\vartheta ) +\delta \vartheta. }$$

   (3.182)

3.5.6 The Entropy Production Rate Represented by a Positive Measure

In accordance with the general ideas discussed in Sect. 1.2, the entropy inequality can be interpreted as a weak formulation of a balance law with the production rate represented by a positive measure. More specifically, writing (3.178) in the form

for any \(\varphi \in C_{c}^{\infty }([0,T) \times \overline{\Omega })\), φ ≥ 0, the left-hand side can be understood as a non-negative linear form defined on the space of smooth functions with compact support in \([0,T) \times \overline{\Omega }\).

Consequently, by means of the classical Riesz representation theorem, there exists a regular, non-negative Borel measure \(\Sigma _{\varepsilon,\delta }\) on the set \([0,T) \times \overline{\Omega }\), that can be trivially extended on the compact set \([0,T] \times \overline{\Omega }\) such that

for any \(\varphi \in C_{c}^{\infty }([0,T) \times \overline{\Omega })\). Moreover,

where we have identified the function \(\sigma _{\varepsilon,\delta } \in L^{1}((0,T) \times \Omega )\) with a non-negative measure, see (1.13)–(1.17) for more details.

The next step in the proof of Theorem 3.1 is to let ɛ → 0 in the approximate system (3.171)–(3.181) in order to eliminate the artificial diffusion term in (3.171) as well as the other ɛ-dependent quantities in the remaining equations. Such a step is not straightforward, as we loose the uniform bound on ∇ _x ϱ; whence compactness of ϱ with respect to the space variable becomes an issue. In particular, the lack of pointwise convergence of the densities has to be taken into account in the proof of pointwise convergence of the approximate temperatures; accordingly, the procedure described in the previous section relating formulas (3.150), (3.151) has to be considerably modified. Apart from these principal new difficulties a number of other rather technical issues has to be addressed. In particular, uniform bounds must be established in order to show that all ɛ-dependent quantities in the approximate continuity equation (3.171), momentum equation (3.174), and energy balance (3.181) vanish in the asymptotic limit ɛ → 0. Similarly, the non-negative quantities appearing in the approximate entropy production rate σ _ɛ, δ are used to obtain uniform bounds in order to eliminate the “artificial” entropy flux r in (3.178).

In order to show pointwise convergence of the approximate temperatures, we take advantage of certain general properties of weak convergence of composed functions expressed conveniently in terms of parameterized (Young) measures (see Sect. 3.6.2). On the other hand, similarly to the recently developed existence theory for compressible viscous fluids, we use the extra regularity properties of the quantity \(\Pi:= p(\varrho,\vartheta ) - (\frac{4} {3}\mu (\vartheta ) +\eta (\vartheta ))\mathrm{div}_{x}\mathbf{u}\) called effective viscous flux in order to establish pointwise convergence of the approximate densities. Such an approach requires a proper description of possible oscillations of the densities provided by the renormalized continuity equation (cf. Sect. 11.19 in Appendix).

3.6.1 Uniform Estimates and Limit in the Approximate Continuity Equation

Let {ϱ _ɛ , u _ɛ , ϑ _ɛ } _{ɛ > 0} be a family of solutions to the approximate system (3.171)–(3.181) constructed in Sect. 3.5. Similarly to Sect. 2.2.3, the total energy balance (3.181), together with the entropy inequality represented through (3.183), give rise to the dissipation balance

where \(\Sigma _{\varepsilon,\delta } \in \mathcal{M}^{+}([0,T] \times \overline{\Omega })\) is the entropy production rate introduced in Sect. 3.5.6, and the “approximate Helmholtz function” \(H_{\delta,\overline{\vartheta }}\) is given through (3.100).

Repeating the arguments used after formula (3.104) we obtain

together with

where, in accordance with (3.180), (3.184), estimate (3.187) further implies

and

Exactly as in Sect. 3.5, the above estimates can be used to deduce that

and

at least for suitable subsequences. Moreover, we have \(\mathbf{u}(t,\cdot ) \in W_{\mathbf{n}}^{1,2}(\Omega; \mathbb{R}^{3})\) for a.a. t ∈ (0, T) in the case of the complete slip boundary conditions, while \(\mathbf{u}(t,\cdot ) \in W_{0}^{1,2}(\Omega; \mathbb{R}^{3})\) for a.a. t ∈ (0, T), if the no-slip boundary conditions are imposed.

Multiplying Eq. (3.171) on ϱ _ɛ and integrating by parts we get

whence, taking (3.191)–(3.193) into account, we can see that

in particular,

As the time derivative ∂ _t ϱ _ɛ can be expressed by means of Eq. (3.171), convergence in (3.191) can be, similarly to (3.118), strengthened to

Relation (3.195), combined with (3.192) and boundedness of the kinetic energy, yields

Thus we conclude that the limit functions ϱ, u satisfy the integral identity

for any test function \(\varphi \in C_{c}^{\infty }([0,T) \times \overline{\Omega })\). Moreover, since the boundary \(\partial \Omega\) is regular (Lipschitz) we can extend continuously the velocity field u outside \(\Omega\) in such a way that the resulting vector field belongs to \(W^{1,2}(\mathbb{R}^{3}; \mathbb{R}^{3})\). (In the case of no-slip boundary conditions one can take trivial extension, where u = 0 outside \(\Omega\).) Accordingly, setting ϱ ≡ 0 in \(\mathbb{R}^{3}\setminus \Omega\) we can assume that ϱ, u solve the equation of continuity

3.6.2 Entropy Balance and Strong Convergence of the Approximate Temperatures

Our principal objective is to show strong (pointwise) convergence of the family {ϑ _ɛ } _{ɛ > 0}. Following the same strategy as in Sect. 3.5.3, we divide the proof into three steps:

1. (i)

   Div-Curl lemma (Proposition 3.3) is applied to show that

   $$\displaystyle{\overline{\varrho s_{\delta }(\varrho,\vartheta )G(\vartheta )} = \overline{\varrho s_{\delta }(\varrho,\vartheta )}\;\overline{G(\vartheta )}}$$

   for any G ∈ W ^1,∞(0, ∞). This relation is reminiscent of formula (3.149); the quantity G playing a role of a cut-off function is necessary because of the low integrability of ϑ. The proof uses the same arguments as in Sect. 3.5.3.

2. (ii)

   Although strong convergence of the densities is no longer available at this stage, we can still show that

   $$\displaystyle{ \overline{b(\varrho )G(\vartheta )} = \overline{b(\varrho )}\;\overline{G(\vartheta )}, }$$

   (3.199)

   where b ∈ C([0, ∞)) ∩ L ^∞((0, ∞)), and G is the same as in the previous step. In order to prove this identity, we use the properties of renormalized solutions to the approximate continuity equation (cf. Sect. 11.19 in Appendix). Very roughly indeed, we can say that possible oscillations in the sequence of approximate densities and temperatures take place in orthogonal directions of the space-time.

3. (iii)

   The simple monotonicity argument used in formula (3.150) has to be replaced by a more sophisticated tool. Here, the desired relation

   $$\displaystyle{s_{M}(\varrho _{\varepsilon },t,x)(G(\vartheta _{\varepsilon }) -\overline{G(\vartheta )}) \rightarrow 0}$$

   is shown to follow directly from (3.199) by means of a general argument borrowed from the theory of parameterized (Young) measures. An elementary alternative proof of this step involving a compactness argument based on the renormalized continuity equation (more precisely on Theorem 11.37 in Appendix) is shown in Sect. 3.7.3.

In the remaining part of this section, we develop the ideas delineated in the above program in a more specific way.

Uniform Estimates Seeing that the sequence {ϱ _ɛ , u _ɛ , ϑ _ɛ } _{ɛ > 0} admits the bounds obtained in (3.188), we infer that {ϑ _ɛ } _{ɛ > 0} satisfies the estimates stated in (3.121)–(3.127), namely

Moreover, relations (3.128), (3.129) imply that

Application of Div-Curl Lemma Now we rewrite the approximate entropy balance (3.183) in the form

to be understood in the weak sense specified in Sects. 1.2 and 3.5.6.

Similarly to Sect. 3.5.3, we intend to apply Div-Curl lemma (Proposition 3.3) to the four-component vector fields

where G is a bounded globally Lipschitz function on [0, ∞).

First observe that the families

are relatively compact in \(W^{-1,s}((0,T) \times \Omega ))\), \(W^{-1,s}((0,T) \times \Omega; \mathbb{R}^{4\times 4})\) for \(s \in [1, \frac{4} {3})\), respectively. Indeed, it is enough to use estimates (3.187), (3.191), (3.193), (3.200), and compactness of the embeddings \(\mathcal{M}^{+}([0,T] \times \overline{\Omega })\hookrightarrow W^{-1,s}((0,T) \times \Omega ))\), \(L^{1}((0,T) \times \Omega ))\hookrightarrow W^{-1,s}((0,T) \times \Omega ))\). Notice that we have, in particular,

as a direct consequence of (3.193).

As the sequence {G(ϑ _ɛ )} _{ɛ > 0} is bounded in \(L^{\infty }((0,T) \times \Omega )\), it is enough to show boundedness of the family {U _ɛ } _{ɛ > 0} in \(L^{\,p}((0,T) \times \Omega; \mathbb{R}^{4})\) for a certain 1 < p < ∞. Combining the arguments already used in (3.144), (3.145) with the bounds (3.191), (3.200), we infer that

while

Finally, following the reasoning of (3.146)–(3.148), we use (3.200) and (3.201) to obtain

Having verified all hypotheses of Proposition 3.3 we conclude that

for any bounded and continuous function G.

Monotonicity of the Entropy and Strong Convergence of the Approximate Temperatures: Application of the Theory of Parametrized (Young) Measures Similarly to Sect. 3.5.3, relation (3.208) can be used to show strong (pointwise) convergence of {ϑ _ɛ } _{ɛ > 0}. Decomposing

we have to show that

for any continuous and increasing G chosen in such a way that all the weak limits exist et least in L ¹. Indeed, relations (3.209) combined with (3.208) imply

yielding, up to a subsequence, the desired conclusion

In order to see (3.209), write

Consequently, the first inequality in (3.209) follows as soon as we can show that

The quantity

may be regarded as a superposition of a Carathéodory function with a weakly convergent sequence. In such a situation, a general argument of the theory of parameterized (Young) measures asserts that (3.212) follows as soon as we show that

for arbitrary smooth and bounded functions b and G (see Theorem 13).

Indeed, if ν _(t, x) ^(ϱ, ϑ), ν _(t, x) ^ϱ and ν _(t, x) ^ϑ are families of parametrized Young measures associated to sequences {(ϱ _ɛ , ϑ _ɛ )} _{ɛ > 0}, {ϱ _ɛ } _{ɛ > 0} and{ϑ _ɛ } _{ɛ > 0}, respectively, then (3.213) implies

This evidently yields a decomposition

where A, B are open subsets in \(\mathbb{R}\). Consequently, for any Carathéodory function ψ(t, x, λ) and a continuous function G(ϑ), such that sequences ψ(⋅ , ⋅ , ϱ _n )G(ϑ _n ) and ψ(⋅ , ⋅ , ϱ _n ), G(ϑ _n ) are weakly convergent in \(L^{1}((0,T) \times \Omega; \mathbb{R}^{2})\) and \(L^{1}((0,T) \times \Omega )\), respectively, we have

which is nothing other than (3.212).

In order to verify (3.213), multiply the approximate continuity equation (3.171) by b′(ϱ)φ, \(\varphi \in C_{c}^{\infty }(\Omega )\), and integrate over \(\Omega\) to obtain

Consequently, the sequence \(\{t\mapsto \int _{\Omega }b(\varrho _{\varepsilon })\varphi \}_{\varepsilon>0}\) is uniformly bounded and equi-continuous in C([0, T]); whence

at least for any smooth function b with bounded second derivative.

Now, we use compactness of the embedding \(L^{\Gamma }(\Omega )\hookrightarrow W^{-1,2}(\Omega )\) to deduce

On the other hand, in accordance with the uniform bounds established in (3.200),

whence (3.213) follows from (3.216), (3.217).

In addition to (3.211), the limit temperature field ϑ is positive a.a. on the set \((0,T) \times \Omega\), more precisely, we have

Indeed, (3.218) follows from the uniform bounds (3.200), the pointwise convergence of {ϑ _ɛ } _{ɛ > 0} established in (3.211), and the property of weak lower semi-continuity of convex functionals (see Theorem 11.27 in Appendix).

Asymptotic Limit in the Entropy Balance At this stage, we are ready to let ɛ → 0 in the approximated entropy equality (3.183).

Using relations (3.200)–(3.211) we obtain, in the same way as in (3.155), (3.156),

Furthermore, in accordance with (3.191), (3.211), we get

Applying Div-Curl Lemma (Proposition 3.3) to the sequence {U _ɛ } _{ɛ > 0} defined in (3.202) and {V _ɛ } _{ɛ > 0},

we deduce

The terms \(\frac{1} {\vartheta _{\varepsilon }} S_{\delta }(\vartheta _{\varepsilon },\mathbf{u}_{\varepsilon }): \nabla \mathbf{u}_{\varepsilon }\), \(\frac{\kappa _{\delta }(\vartheta _{\varepsilon })} {\vartheta _{\varepsilon }} \vert \nabla \vartheta _{\varepsilon }\vert ^{2}\) appearing in σ _ɛ, δ are weakly lower semi-continuous as we have already observed in (3.159)–(3.164), while the remaining ɛ-dependent quantities in σ _ɛ, δ are non-negative. Finally, by virtue of (3.187), we may assume

Recalling the limits (3.204) and (3.207), we let ɛ → 0 in (3.183) to obtain

where

Consequently, in order to perform the limit ɛ → 0 in the remaining equations of the approximate system (3.171)–(3.181), we have to show

1. (i)

   uniform pressure estimates analogous to those established in Sect. 2.2.5 or, alternatively, in Sect. 2.2.6,

2. (ii)

   strong (pointwise) convergence of the approximate densities.

3.6.3 Uniform Pressure Estimates

The pressure estimates are derived in the same way as in Sect. 2.2.5, namely we use the quantities

as test functions in the approximate momentum equation (3.174), where

and \(\mathcal{B}\approx \mathrm{div}_{x}^{-1}\) is the Bogovskii operator introduced in Sect. 2.2.5 and investigated in Sect. 11.6 in Appendix.

Since ϱ _ɛ satisfies the approximate continuity equation (3.171), we have

Consequently, by virtue of the basic properties of the operator \(\mathcal{B}\) listed in Sect. 2.2.5,

and

for any 1 < p < ∞.

The last two estimates, together with those previously established in (3.191)–(3.196), (3.200), and (3.201), render the test functions (3.220) admissible in (3.174) provided, say, \(\Gamma \geq 4\). Note that, unlike in Sect. 2.2.5, the argument of the operator \(\mathcal{B}\) is an affine function of ϱ _ɛ , whereas the necessary uniform estimate on {ϱ _ɛ } _{ɛ > 0} in \(L^{\infty }(0,T;L^{\Gamma }(\Omega ))\) is provided by the extra pressure term \(\delta \varrho ^{\Gamma }\).

In view of these arguments, we can write, similarly to (2.94),

where

and

The simple form of I ₇ conditioned by the specific form of the test function φ, where the argument of \(\mathcal{B}\) is an affine function of ϱ _ɛ , is the only technical reason why the limit processes for ɛ → 0 and δ → 0 must be separated.

The integral identity (3.224) can be used to obtain uniform bounds on the pressure independent of ɛ. Exactly as in Sect. 2.2.5, we deduce that

and

Indeed, these bounds can be obtained by dominating the integrals I ₁ − I ₇ in the spirit of Sect. 2.2.5, specifically, by means of estimates (3.222), (3.223), (3.191)–(3.196), and (3.200), provided \(\Gamma \geq 4\). In particular, by virtue of (3.192), (3.194),

yielding boundedness of integral I ₇.

3.6.4 Limit in the Approximate Momentum Equation and in the Energy Balance

In accordance with estimates (3.225), (3.226), together with (3.193), (3.200), and (3.211),

where we have denoted

On the other hand, by virtue of (3.17), (3.23), (3.193), and (3.211),

Moreover, since \(\mathbb{S}_{\delta }\) takes the form specified in (3.53), we can use (3.192) in order to deduce

As the limits of the families ϱ _ɛ f, ϱ _ɛ u _ɛ , and ɛ∇ϱ _ɛ ∇u _ɛ have already been identified through (3.191), (3.196) and (3.227), we are left with the convective term ϱ _ɛ u _ɛ ⊗ u _ɛ . Following the arguments of Sect. 3.5.4 we observe that

Consequently, because of compact embedding \(L^{s}(\Omega )\hookrightarrow W^{-1,2}(\Omega )\), \(s> \frac{6} {5}\),

for any 1 ≤ p < ∞. In accordance with (3.192),

Letting ɛ → 0 in the approximate momentum equation (3.174) we get

for any test function \(\boldsymbol{\varphi }\in C_{c}^{\infty }([0,T) \times \overline{\Omega };R^{3}))\) such that either

or

Finally, as the sequence {ϱ _ɛ e _δ (ϱ _ɛ , ϑ _ɛ )} _{ɛ > 0} is bounded in \(L^{\,p}((0,T) \times \Omega )\) (see (3.30), (3.191)–(3.193), (3.200)), we are allowed to let ɛ → 0 in the approximate energy balance (3.181) to obtain

3.6.5 Strong Convergence of the Densities

In order to show strong (pointwise) convergence of {ϱ _ɛ } _{ɛ > 0}, we adapt the method introduced in the context of barotropic fluids with constant viscosity coefficients by Lions [192], and further developed in [103] in order to accommodate the variable transport coefficients.

Similarly to Sect. 2.2.6, we use the quantities

as test functions in the approximate momentum equation (3.174), where the symbol \(\Delta _{x}^{-1}\) stands for the inverse Laplace operator considered on the whole space \(\mathbb{R}^{3}\) introduced in (2.100). The operator \(\nabla _{x}\Delta _{x}^{-1}\) is investigated in Sect. 11.17 in Appendix.

Since ϱ _ɛ u _ɛ and ∇ϱ _ɛ possess zero normal traces, the approximate continuity equation (3.171) can be extended to the whole \(\mathbb{R}^{3}\), specifically,

Accordingly, we have

cf. Theorem 11.33 in Appendix.

Now, exactly as in Sect. 2.2.6, we can use the uniform estimates (3.191)–(3.196), (3.200), and (3.201), in order to observe that φ defined through (3.235) is admissible in the integral identity (3.174) as soon as \(\Gamma \geq 4\). Thus we get

where

and

Here, the symbol \(\mathcal{R}\) stands for the double Riesz transform, defined componentwise as \(\mathcal{R}_{i,j} = \partial _{x_{i}}\Delta _{x}^{-1}\partial _{x_{j}}\), introduced in (2.101).

Repeating the same procedure we use the quantities

as test functions in the limit momentum equation (3.233) in order to obtain

where

and

Combining (3.191) with (3.215) we get

In accordance with the standard theory of elliptic problems, the pseudodifferential operator \((\nabla _{x}\Delta _{x}^{-1})\) “gains” one spatial derivative, in particular, by virtue of the embedding \(W^{1,\Gamma }(\Omega )\hookrightarrow C(\overline{\Omega })\), we get

provided \(\Gamma> 3\) (see Theorem 11.33 in Appendix). Consequently, we can use relations (3.191), (3.196), (3.228)–(3.232) in order to see that (i) I _2,ɛ , I _8,ɛ → 0, while (ii) the integrals I _j, ɛ , j = 3, …, 7, converge for ɛ → 0 to their counterparts in (3.239). We infer that

Now, the crucial observation is that the difference of the two most right quantities in (3.240) vanishes. In order to see this, we need the following assertion (Theorem 11.34 in Appendix) that can be viewed as a straightforward consequence of the Div-Curl lemma.

Lemma 3.5

Let

where

Then

This statement provides the following corollary:

Corollary 3.3

Let

where

Then

Hereafter, we shall use Corollary 3.3 to show that

where, recall, \(\mathcal{R}[\mathbf{v}]\) is a vector field with i-th component \(\sum _{j=1}^{3}\mathcal{R}_{i,j}[v_{j}]\) while \(\mathcal{R}[a]\mathbf{v}\) is a vector field with i-th component \(\sum _{j=1}^{3}\mathcal{R}_{i,j}[a]v_{j}\).

As shown in (3.195), (3.231),

Applying Corollary 3.3 to r _ɛ = ϱ _ɛ (t, ⋅ ), U _ɛ = ϱ _ɛ u _ɛ (t, ⋅ ) (extended by 0 outside \(\Omega\)), we obtain

for all t ∈ [0, T].

As the embedding \(L^{ \frac{2\Gamma } {\Gamma +3} }(\Omega )\hookrightarrow W^{-1,2}(\Omega )\) is compact for \(\Gamma> 9/2\), we conclude that

which, together with (3.192), yields (3.241). Consequently, (3.240) reduces to

Our next goal is to replace in (3.243) the quantity \(\mathbb{S}_{\delta }(\vartheta _{\varepsilon },\nabla _{x}\mathbf{u}_{\varepsilon }): \mathcal{R}[1_{\Omega }\varrho _{\varepsilon }]\) by \(\varrho _{\varepsilon }\Big(\frac{4} {3}\mu _{\delta }(\vartheta _{\varepsilon })\) \(+\eta (\vartheta _{\varepsilon })\Big)\mathrm{div}_{x}\mathbf{u}_{\varepsilon }\), and, similarly, \(\mathbb{S}_{\delta }(\vartheta,\nabla _{x}\mathbf{u}): \mathcal{R}[1_{\Omega }\varrho ]\) by the expression \(\varrho \Big(\frac{4} {3}\mu _{\delta }(\vartheta )\) \(+\eta (\vartheta )\Big)\mathrm{div}_{x}\mathbf{u}\) in (3.243), where μ _δ (ϑ) = μ(ϑ) + δϑ.

To this end write

and

where we have used the evident properties of the double Riesz transform recalled in Sect. 11.17 in Appendix. Furthermore,

and

with the commutator

In order to proceed, we report the following result in the spirit of Coifman and Meyer [62] proved as Theorem 11.35 in Appendix.

■      Commutator Lemma:

Lemma 3.6

Let \(w \in W^{1,2}(\mathbb{R}^{3})\) and \(\mathbf{Z} \in L^{\,p}(\mathbb{R}^{3}; \mathbb{R}^{3})\) be given, where \(\frac{6} {5} <p <\infty\) .

Then for any \(1 <s <\frac{6p} {6+p}\) ,

where \(0 <\beta = \frac{3} {s} -\frac{6+p} {6p} <1\) and c = c( p, s) is a positive constant.

Applying Lemma 3.6 to w = w _ɛ = ζ(μ(ϑ _ɛ ) + δϑ _ɛ ), Z = Z _ɛ = [Z _ɛ, 1, Z _ɛ, 2, Z _ɛ, 3], with \(Z_{\varepsilon,i} = \partial _{x_{i}}u_{\varepsilon,j} + \partial _{x_{j}}u_{\varepsilon,i}\), j = 1, 2, 3, where {w _ɛ } _{ɛ > 0}, {Z _ɛ } _{ɛ > 0} are bounded in \(L^{2}(0,T;W^{1,2}(\Omega ))\) and \(L^{2}((0,T) \times \Omega; \mathbb{R}^{3})\), respectively, cf. (3.192), (3.200), (3.17)–(3.18), we deduce that

for certain \(1 <s <\frac{3} {2}\), \(0 <\beta = \frac{3-2s} {s} <1\).

Next we claim that

where, in accordance with relations (3.17), (3.23), (3.192), (3.193), and (3.211),

In order to show (3.245), we apply the Div-Curl Lemma (see Proposition 3.3) to the four-component vector fields

In view of relations (3.171), (3.194), (3.244) yielding the sequences {div _t, x U _ɛ } _{ɛ > 0} and {curl _t, x V _ɛ } _{ɛ > 0} compact in \(W^{-1,s}((0,T) \times \Omega )\) and \(W^{-1,s}((0,T) \times \Omega; \mathbb{R}^{3\times 3})\) for a certain s > 1, it is enough to observe that

with 1∕r + 1∕q < 1. This is certainly true provided \(\Gamma\) is large enough.

Relations (3.243), (3.245), (3.246) give rise to a remarkable identity

■     Weak Compactness Identity for Effective Pressure (Level ɛ):

where the quantity \(p - (\frac{4} {3}\mu +\eta )\mathrm{div}_{x}\mathbf{u}\) is usually termed effective viscous flux or effective pressure. As we will see below, the quantity

plays a role of a “defect” measure of the density oscillations described through the (renormalized) equation of continuity. Relation (3.247) enables us to relate these oscillations to the changes in the pressure.

In order to exploit (3.247), we multiply the approximate continuity equation (3.171) on G′(ϱ _ɛ ), where G is a smooth convex function, integrate by parts, and let ɛ → 0 to obtain

from which we easily deduce that

for a.a. τ ∈ (0, T).

To derive a relation similar to (3.249) for the limit functions ϱ, u, we need the renormalized continuity equation introduced in (1.20). Note that we have already shown that the quantities ϱ, u solve the continuity equation (3.197) in \((0,T) \times \mathbb{R}^{3}\). On the other hand, the general theory of transport equations developed by DiPerna-Lions asserts that any solution of (3.197) is automatically a renormalized one as soon as, roughly speaking, the quantity ϱdiv _x u is integrable.

More precisely, we report the following result proved in Sect. 11.19 in Appendix.

Lemma 3.7

Assume that \(\varrho \in L^{2}((0,T) \times \mathbb{R}^{3})\) , \(\mathbf{u} \in L^{2}(0,T;W^{1,2}(\mathbb{R}^{3}))\) solve the equation of continuity ( 3.197 ) in \(\mathcal{D}'((0,T) \times \mathbb{R}^{3}))\) .

Then ϱ, u represent a renormalized solution in the sense specified in ( 2.2 ).

As a consequence of Lemma 3.7 (see also Theorem 11.36 and Lemma 11.13 for more details), we deduce

Since the pressure p _δ is non-decreasing with respect to ϱ and we already know that ϑ _ɛ → ϑ strongly in \(L^{1}((0,T) \times \Omega )\), we have

Indeed,

where the first term is non-negative, and the second term tends to zero by virtue of the asymptotic limits established in (3.191), (3.211), the bounds (3.193), (3.200), (3.225), (3.226), and the structural properties of p _δ stated in (3.229).

Consequently, relation (3.247) yields

whence (3.249) together with (3.250) imply the desired conclusion

As z ↦ zlog(z) is a strictly convex function, we may infer that

in agreement with Theorem 11.27 in Appendix.

3.6.6 Artificial Diffusion Asymptotic Limit

Strong convergence of the sequence of approximate densities established in (3.251) completes the second step in the proof of Theorem 3.1 eliminating completely the ɛ-dependent terms in the approximate system. For any δ > 0, we have constructed a trio {ϱ, u, ϑ} solving the following problem:

1. (i)

   Renormalized continuity equation:

   $$\displaystyle{ \int _{0}^{T}\int _{ \Omega }\varrho B(\varrho )\Big(\partial _{t}\varphi + \mathbf{u} \cdot \nabla _{x}\varphi \Big)\ \mathrm{d}x\ \mathrm{d}t }$$

   (3.252)
   $$\displaystyle{=\int _{ 0}^{T}\int _{ \Omega }b(\varrho )\mathrm{div}_{x}\mathbf{u}\varphi \ \mathrm{d}x\ \mathrm{d}t -\int _{\Omega }\varrho _{0,\delta }B(\varrho _{0,\delta })\varphi (0,\cdot )\ \mathrm{d}x}$$

   for any

   $$\displaystyle{b \in L^{\infty }\cap C[0,\infty ),\ B(\varrho ) = B(1) +\int _{ 1}^{\varrho }\frac{b(z)} {z^{2}} \ \mathrm{d}z,}$$

   and any test function

   $$\displaystyle{\varphi \in C_{c}^{\infty }([0,T) \times \overline{\Omega }).}$$
2. (ii)

   Approximate balance of momentum:

   $$\displaystyle{ \int _{0}^{T}\int _{ \Omega }\Big(\varrho \mathbf{u} \cdot \partial _{t}\boldsymbol{\varphi } +\varrho [\mathbf{u} \otimes \mathbf{u}]: \nabla _{x}\boldsymbol{\varphi } +\Big (\,p +\delta (\varrho ^{\Gamma } +\varrho ^{2})\Big)\mathrm{div}_{ x}\boldsymbol{\varphi }\Big)\ \mathrm{d}x\ \mathrm{d}t }$$

   (3.253)
   $$\displaystyle{=\int _{ 0}^{T}\int _{ \Omega }\Big(\mathbb{S}_{\delta }: \nabla _{x}\boldsymbol{\varphi } -\varrho \mathbf{f}_{\delta }\cdot \boldsymbol{\varphi }\Big)\ \mathrm{d}x\ \mathrm{d}t -\int _{\Omega }(\varrho \mathbf{u})_{0}\cdot \boldsymbol{\varphi }\ \mathrm{d}x,}$$

   for any test function \(\boldsymbol{\varphi }\in C_{c}^{\infty }([0,T) \times \overline{\Omega }; \mathbb{R}^{3})\), where either

   $$\displaystyle{ \boldsymbol{\varphi }\cdot \mathbf{n}\vert _{\partial \Omega } = 0\ \mbox{ in the case of the complete slip boundary conditions,} }$$

   (3.254)

   or

   $$\displaystyle{ \boldsymbol{\varphi }\vert _{\partial \Omega } = 0\ \mbox{ in the case of the no-slip boundary conditions.} }$$

   (3.255)

   Furthermore,

   $$\displaystyle{ \mathbb{S}_{\delta } = (\mu (\vartheta )+\delta \vartheta )\Big(\nabla _{x}\mathbf{u} + \nabla _{x}^{\perp }\mathbf{u} -\frac{2} {3}\mathrm{div}_{x}\mathbf{u}\ \mathbb{I}\Big) +\eta (\vartheta )\mathrm{div}_{x}\mathbf{u}\ \mathbb{I}. }$$

   (3.256)
3. (iii)

   Approximate entropy balance:

   $$\displaystyle{ \int _{0}^{T}\int _{ \Omega }\varrho s_{\delta }(\varrho,\vartheta )\Big(\partial _{t}\varphi + \mathbf{u} \cdot \nabla _{x}\varphi \Big)\ \mathrm{d}x\ \mathrm{d}t }$$

   (3.257)
   $$\displaystyle{+\int _{0}^{T}\int _{ \Omega }\frac{\kappa _{\delta }(\vartheta )\nabla _{x}\vartheta } {\vartheta } \cdot \nabla _{x}\varphi \ \mathrm{d}x\ \mathrm{d}t+ <\sigma _{\delta };\varphi> _{[\mathcal{M},C]([0,T]\times \overline{\Omega })} =}$$
   $$\displaystyle{-\int _{\Omega }(\varrho s)_{0,\delta }\varphi (0,\cdot )\ \mathrm{d}x -\int _{0}^{T}\int _{ \Omega }\frac{\varrho } {\vartheta }\mathcal{Q}_{\delta }\varphi \ \mathrm{d}x\ \mathrm{d}t}$$

   for all \(\varphi \in C_{c}^{\infty }([0,T) \times \overline{\Omega })\), where \(\sigma _{\delta } \in \mathcal{M}^{+}([0,T] \times \overline{\Omega })\) satisfies

   $$\displaystyle{ \sigma _{\delta } \geq \frac{1} {\vartheta } \Big[\mathbb{S}_{\delta }: \nabla _{x}\mathbf{u} +\Big (\frac{\kappa (\vartheta )} {\vartheta } + \frac{\delta } {2}(\vartheta ^{\Gamma -1} + \frac{1} {\vartheta ^{2}} )\Big)\vert \nabla _{x}\vartheta \vert ^{2} +\delta \frac{1} {\vartheta ^{2}} \Big], }$$

   (3.258)

   and where we have set

   $$\displaystyle{ s_{\delta }(\varrho,\vartheta ) = s(\varrho,\vartheta ) +\delta \log (\vartheta ),\ \kappa _{\delta }(\vartheta ) =\kappa (\vartheta ) +\delta \Big (\vartheta ^{\Gamma } + \frac{1} {\vartheta } \Big). }$$

   (3.259)
4. (iv)

   Approximate energy balance:

   $$\displaystyle{ \int _{\Omega }\Big(\frac{1} {2}\varrho \vert \mathbf{u}\vert ^{2} +\varrho e(\varrho,\vartheta ) + \delta ( \frac{\varrho ^{\Gamma }} {\Gamma - 1} +\varrho ^{2})\Big)(\tau )\ \mathrm{d}x }$$

   (3.260)
   $$\displaystyle{= \int _{\Omega }\Big(\frac{1} {2} \frac{\vert (\varrho \mathbf{u})_{0}\vert ^{2}} {\varrho _{0,\delta }} +\varrho _{0,\delta }e_{0,\delta } + \delta ( \frac{\varrho _{0,\delta }^{\Gamma }} {\Gamma - 1} +\varrho _{ 0,\delta }^{2})\Big)\ \mathrm{d}x}$$
   $$\displaystyle{+\int _{0}^{\tau }\int _{ \Omega }\Big(\varrho \mathbf{f}_{\delta } \cdot \mathbf{u} +\varrho \mathcal{Q}_{\delta } +\delta \frac{1} {\vartheta ^{2}} \Big)\ \mathrm{d}x\ \mathrm{d}t\ \mbox{ for a.a.}\ \tau \in [0,T].}$$

The last and probably the most illuminative step in the proof of Theorem 3.1 is to let δ → 0 in the approximate system (3.252)–(3.260). Although many arguments are almost identical or mimick closely those discussed in the previous text, there are still some new ingredients coming into play. Notably, we introduce a concept of oscillation defect measure in order to control the density oscillations beyond the theory of DiPerna and Lions. Moreover, weighted estimates of this quantity are used in order to accommodate the physically realistic growth restrictions on the transport coefficients imposed through hypotheses (3.17), (3.23).

3.7.1 Uniform Estimates

From now on, let {ϱ _δ , u _δ , ϑ _δ } _{δ > 0} be a family of approximate solutions satisfying (3.252)–(3.260). To begin, we recall that the total mass is a constant of motion, specifically,

Since we assume that

the bound (3.261) is uniform for δ → 0.

The next step is the dissipation balance

satisfied for a.a. τ ∈ [0, T], which can be deduced from (3.257), (3.260), with the Helmholtz function \(H_{\overline{\vartheta }}\) introduced in (2.48). Accordingly, in order to get uniform estimates, we have to take

as well as

uniformly for δ → 0.

As the term δ∕ϑ _δ ² is “absorbed” by its counterpart in the entropy production σ _δ satisfying (3.258), the dissipation balance (3.263) gives rise, exactly as in Sect. 3.6.1, to the following uniform estimates:

and

In addition, we have

and, as a consequence of (3.258),

and

Finally, making use of Korn’s inequality established in Proposition 2.1 we deduce, exactly as in (2.65), (2.66), that

and

where α was introduced in hypotheses (3.17)–(3.23). Moreover,

Note that estimates (3.269)–(3.272) yield

while (3.275), (3.276), together with hypotheses (3.17), (3.19), and (3.23), imply that

(cf. estimate (2.68)).

Now, positivity of the absolute temperature can be shown by help of Proposition 2.2 and Lemma 2.1, exactly as in Sect. 2.2.4. In particular, estimate (3.271) can be strengthened to

In order to complete our list of uniform bounds, we evoke the pressure estimates obtained in Sect. 2.2.5. In the present context, relation (2.95) reads

where ν > 0 is a constant exponent.

3.7.2 Asymptotic Limit for Vanishing Artificial Pressure

The piece of information provided by the uniform bounds established in the previous section is sufficient for taking δ → 0 in the approximate system of equations (3.252)–(3.260).

Due to the structural properties of the molecular pressure p _M derived in (3.32), and because of (3.229), estimates (3.267), (3.269), and (3.275), (3.276) imply that

and

at least for suitable subsequences.

Taking b ≡ 0 in the renormalized Eq. (3.252), we deduce, in view of the previous estimates, that

On the other hand, as the Lebesgue space \(L^{\frac{5} {3} }(\Omega )\) is compactly embedded into the dual \(W^{-1,p'}(\Omega )\), p′ = 8∕(3 + α) as soon as α ∈ (2∕5, 1], we conclude, taking (3.282) together with (3.266), (3.267) into account, that

A similar argument in the case when the time derivative of the momentum ϱ _δ u _δ is expressed via the approximate momentum equation (3.253) gives rise to

Since

we can use (3.284) to conclude that

In order to handle the approximate pressure in the momentum equation (3.253), we first observe that, as a direct consequence of (3.281),

Moreover, writing

and interpolating estimates (3.269), (3.278), we have

In accordance with hypotheses (3.15), (3.16), the asymptotic structure of p _M derived in (3.32), and in agreement with (3.281), (3.291),

At this stage, it is possible to let δ → 0 in Eqs. (3.252), (3.253) to obtain

for any test function \(\varphi \in C_{c}^{\infty }([0,T) \times \overline{\Omega })\) and any

Similarly, we get

for any test function \(\boldsymbol{\varphi }\in C_{c}^{\infty }([0,T) \times \overline{\Omega }; \mathbb{R}^{3})\) satisfying \(\boldsymbol{\varphi }\cdot \mathbf{n}\vert _{\partial \Omega } = 0\), or, in addition, \(\boldsymbol{\varphi }\vert _{\partial \Omega } = 0\) in the case of the no-slip boundary conditions. Here we have set

Finally, letting δ → 0 in the approximate total energy balance (3.260) we conclude

where we have used estimate (3.273) in order to eliminate the singular term δ∕ϑ _δ ². Moreover, we have assumed strong convergence (a.a.) of the approximate data f _δ , ϱ _0,δ , ϑ _0,δ , and \(\mathcal{Q}_{\delta }\).

3.7.3 Entropy Balance and Pointwise Convergence of the Temperature

Similarly to the preceding parts, specifically Sect. 3.6.2, our aim is to use Div-Curl lemma (Proposition 3.3), together with the monotonicity of the entropy, in order to show

Uniform Estimates We have to show that all terms appearing on the left-hand side of the approximate entropy balance (3.257) are either non-negative or belong to an L ^p-space, with p > 1.

To this end, we use the structural properties of the specific entropy s stated in (3.34), (3.39), together with the uniform estimates (3.267), (3.269), (3.280), to deduce that

Similarly, we have

whence, by virtue of (3.288), combined with estimates (3.282)–(3.284), there is p > 1 such that

In addition, relations (3.277), (3.290) give rise to

The entropy flux can be handled by means of the uniform estimates established in (3.269), (3.278). Indeed, writing

we observe easily that

for a suitable p > 1.

Finally, relations (3.269), (3.274), (3.280) can be used to obtain

uniformly for δ → 0. Consequently, seeing that

we can use (3.283), (3.302), together with Hölder’s inequality and the embedding relation \(W^{1,2}(\Omega )\hookrightarrow L^{6}(\Omega )\), in order to conclude that

Similarly, by the same token,

Strong Convergence of Temperature via the Young Measures Having established all necessary estimates we can proceed as in Sect. 3.6.2.

By virtue of (3.280),

We can apply Div-Curl lemma (Proposition 3.3) in order to obtain identity

Consequently, employing Theorem 11.37, we show identity (3.213). Now we apply Theorem 13 in the same way as in Sect. 3.6.2 and conclude that

We also observe that, according to Theorem 11.26,

The symbol G in the last four formulas denotes an arbitrary nondecreasing and continuous function on [0, ∞), chosen in such a way that all the L ¹-weak limits in the above formulas exist.

Relations (3.307)–(3.308) combined with identity (3.306) yield (3.210). The latter identity implies the pointwise convergence (3.297).

Strong Convergence of Temperature: An Alternative Proof The departure point is formula (3.306) with G(ϑ) = T _k (ϑ), where the truncation functions T _k is defined by formula (3.316) below. The goal is to show the inequality (3.307) by using more elementary arguments than in the previous section. Once this is done, (3.306) and Theorem 11.26 yield

Since the sequence ϑ _δ is bounded in \(L^{\infty }(0,T;L^{4}(\Omega )) \cap L^{2}(0,T;L^{6}(\Omega ))\), the last inequality and Corollary 11.2 in Appendix, imply

which proves (3.297).

Accordingly, it is enough to show

Due to Corollary 11.2 and property (3.39), we have

and

as k → ∞. It is therefore sufficient to prove

Due to the monotonicity of function ϑ ↦ s _M (ϱ, ϑ), we have

Therefore, (3.310) will be verified if we show that

where B is an arbitrary ball in \((0,T) \times \Omega\).

Since log is a concave function, we have \(\overline{\log (T_{k}(\vartheta ))} \leq \log (\overline{T_{k}(\vartheta )})\). Moreover, the sequence {log(ϑ _δ )} _{δ > 0} is bounded in \(L^{2}(0,T;W^{1,2}(\Omega ))\) and the same holds for {log(T _k (ϑ _δ ))} _{δ > 0}. Consequently,

therefore \(\log (\overline{T_{k}(\vartheta )})\) belongs to the space \(L^{2}(0,T;L^{6}(\Omega ))\). In particular, there exists \(z_{\epsilon } \in C^{1}([0,T] \times \overline{\Omega })\) such that

where ɛ > 0 is a parameter that can be taken arbitrarily small. Setting \(\Theta =\exp (z_{\varepsilon })\) we have

Now, we write

We may use (3.11), (3.34) to verify that

Since ϱ _δ is bounded in \(L^{\infty }(0,T;L^{\frac{5} {3} }(\Omega ))\), we infer that

whence the first integral on the right-hand side of (3.311) tends to 0 as ɛ → 0+.

As a consequence of (3.39), the sequence \(B(t,x,\varrho _{\delta }) =\varrho _{\delta }s_{M}(\varrho _{\delta },\Theta (t,x))\) satisfies hypothesis (11.131) of Theorem 11.37 in Appendix. We can therefore conclude that

which, together with the fact that \(T_{k}(\vartheta _{\delta }) \rightarrow \overline{T_{k}(\vartheta )}\) weakly in \(L^{2}(0,T;W^{1,2}(\Omega ))\), concludes the proof of inequality (3.309).

Asymptotic Limit in the Entropy Balance Using weak lower semicontinuity of convex functionals, we can let δ → 0 in the approximate entropy balance (3.257) to conclude that

for any \(\varphi \in C_{c}^{\infty }([0,T) \times \overline{\Omega })\). In this equation

and \(\sigma \in \mathcal{M}^{+}([0,T] \times \overline{\Omega })\) is a weak-(*) limit in \(\mathcal{M}([0,T] \times \overline{\Omega })\) of the sequence σ _δ that exists at least for a chosen subsequence due to estimate (3.270). Employing (3.258), (3.270), (3.284), (3.297) and lower weak semicontinuity of convex functionals, using the fact that all δ-dependent quantities in the entropy production rate at the right hand side of (3.258) are non negative, we show that

For more details see the similar reasoning between formulas (3.158)–(3.160) in Sect. 3.5.3.

Consequently, in order to complete the proof of Theorem 3.1, we have to show pointwise convergence of the densities. This will be done in the next section.

3.7.4 Pointwise Convergence of the Densities

We follow the same strategy as in Sect. 3.6.5, however, some essential steps have to be considerably modified due to lower L ^p-integrability available for {ϱ _δ } _{δ > 0}, {u _δ } _{δ > 0}.

To begin, we introduce a family of cut-off functions

where T ∈ C ^∞[0, ∞),

Similarly to Sects. 2.2.6, 3.6.5, we use the quantities

with the operators \((\nabla _{x}\Delta _{x}^{-1})\) introduced in (2.100), as test functions in the approximate momentum equation (3.253) to deduce

where \(\mathbb{S}_{\delta }:= \mathbb{S}_{\delta }(\vartheta _{\delta },\nabla _{x}\mathbf{u}_{\delta })\) and

and

Now, mimicking the strategy of Sect. 3.6.5, we use

as test functions in the limit momentum balance (3.294) to obtain

where

and

We recall that \(\mathcal{R} = \mathcal{R}_{i,j}\) is the double Riesz transform introduced in Sect. 5.

To get formula (3.319) we have used (3.284), (3.297) to identify \(\overline{\vartheta ^{4}}\) with ϑ ⁴ and \(\overline{\mathbb{S}(\vartheta,\nabla _{x}\mathbf{u})}\) with \(\mathbb{S}:= \mathbb{S}(\vartheta,\nabla _{x}\mathbf{u})\). We also recall that \(\mathcal{R} = \mathcal{R}_{i,j}\) is the double Riesz transform introduced in Sect. 5.

Now, letting δ → 0+ in (3.318), we get

where the right hand side is the same as the right hand side in (3.319). Here, we have used the commutator lemma in form of Corollary 3.3 with \(r_{\delta } = 1_{\Omega }\,T_{k}(\varrho _{\delta })\) and \(\mathbf{V}_{\delta } = 1_{\Omega }\varrho _{\delta }\mathbf{u}_{\delta }\) to show that

exactly in the same way as explained in detail in Sect. 3.6.5. We have also employed the pointwise convergence (3.297) to verify that \(\overline{\vartheta ^{4}} =\vartheta ^{4}\) and that \(\overline{\vartheta ^{4}\,T_{k}(\varrho )} =\) \(\vartheta ^{4}\overline{T_{k}(\varrho )}\).

Combining (3.319) and (3.320), we get identity

We again follow the great lines of Sect. 3.6.5. Employing the evident properties of the Riesz transform evoked in formulas (11.107), we may write

and

where

Applying Lemma 3.6 to w = ζμ(ϑ _δ ), \(\mathbf{Z} = [\partial _{x_{i}}u_{\varepsilon,j} + \partial _{x_{j}}u_{\varepsilon,i}]_{i=1}^{3}\), j ∈ {1, 2, 3} fixed, where, according to (3.17)–(3.18), (3.278), (3.275), the sequences w, Z are bounded in \(L^{2}(0,T;W^{1,2}(\Omega ))\) and \(L^{8/(5-\alpha )}((0,T) \times \Omega )\), respectively, we deduce that

Now, we consider four-dimensional vector fields

and take advantage of relations (3.252), (3.266), (3.267), (3.269), (3.278), (3.275) and (3.321) in order to show that U _δ , V _δ verify all hypotheses of the Div-Curl lemma stated in Proposition 3.3. Using this proposition, we may conclude that

where we have used (3.284), (3.297) to identify \(\overline{\omega (\vartheta,\mathbf{u})}\) with ω(ϑ, u).

We thus discover on this level of approximations again the weak compactness identity for the effective pressure

■      Weak Compactness Identity for Effective Pressure (Level δ):

Thus our ultimate goal is to use relation (3.323) in order to show pointwise convergence of the family of approximate densities {ϱ _δ } _{δ > 0}. To this end, we revoke the “renormalized” limit Eq. (3.293) yielding

for any test function \(\varphi \in C_{c}^{\infty }([0,T) \times \overline{\Omega })\), where we have set

Assume, for a moment, that the limit functions ϱ, u also satisfy the equation of continuity in the sense of renormalized solutions, in particular,

for any test function \(\varphi \in C_{c}^{\infty }([0,T) \times \overline{\Omega })\).

Now, relations (3.324), (3.325) give rise to

As {ϑ _δ } _{δ > 0} converges strongly in L ¹ and p _M is a non-decreasing function of ϱ, we can use relation (3.323) to obtain

Letting k → ∞ in (3.326) we obtain

as soon as we are able to show that

Relation (3.327) yields

see Theorem 11.27 in Appendix. This completes the proof of Theorem 3.1.

Note, however, that two fundamental issues have been left open in the preceding discussion, namely

• the validity of the renormalized Eq. (3.325),

• relation (3.328).

These two intimately related questions will be addressed in the following section.

3.7.5 Oscillations Defect Measure

The oscillations defect measure introduced in [117] represents a basic tool for studying density oscillations. Given a family {ϱ _δ } _{δ > 0}, a set Q, and q ≥ 1, we introduce:

■      Oscillations Defect Measure:

where T _k are the cut-off functions introduced in (3.316).

Assume that

Seeing that

we conclude easily that (3.331) implies (3.328).

A less obvious statement is the following assertion.

Lemma 3.8

Let \(Q \subset \mathbb{R}^{4}\) be an open set. Suppose that

and

where ϱ _δ , u _δ solve the renormalized Eq. ( 2.2 ) in \(\mathcal{D}'(Q)\) .

Then the limit functions ϱ, u solve the renormalized Eq. ( 2.2 ) in \(\mathcal{D}'(Q)\) .

Proof

Clearly, it is enough to show the result on the set J × K, where J is a bounded time interval and K is a ball such that \(\overline{J \times K} \subset Q\). Since ϱ _δ is a renormalized solution of (2.2), we get

whence, by virtue of hypotheses (3.332), (3.333),

Consequently, we deduce

Since \(\overline{T_{k}(\varrho )}\) are bounded, we can apply the regularization technique introduced by DiPerna and Lions [85] (Theorem 11.36), already used in Lemma 3.7, in order to deduce

where h is a continuously differentiable function such that h′(z) = 0 for all z large enough, say, z ≥ M.

Consequently, it is enough to show

To this end, denote

Consequently,

where 1∕r + 1∕r′ = 1.

Furthermore, a simple interpolation argument yields

with β ∈ (0, 1).

As the family {ϱ _δ } _{δ > 0} is equi-integrable, we deduce

Finally, seeing that | T _k ^′(ϱ _δ )ϱ _δ | ≤ T _k (ϱ _δ ), we conclude

whence

Clearly, relation (3.336) follows from (3.337) to (3.340).

□

In order to apply Lemma 3.8, we need to establish suitable bounds on osc _q [ϱ _δ → ϱ]. To this end, revoking (3.42)–(3.44) we write

where

and

for all ϱ, ϑ > 0.

Consequently,

for any \(\varphi \in C_{c}^{\infty }((0,T) \times \Omega )\), φ ≥ 0, where we have used (3.342), convexity of \(\varrho \mapsto \varrho ^{\frac{5} {3} }\), and concavity of T _k on [0, ∞).

In accordance with the uniform bound (3.269) and (3.343), we have

Furthermore, introducing a Carathéodory function

we get, in accordance with (3.344),

Thus, evoking once more (3.323) we infer that

On the other hand, by virtue of hypothesis (3.17) and estimate (3.275), we get

Taking

and using Hölder’s inequality, we obtain

Finally, choosing q such that

we can combine relations (3.346), (3.347), together with estimate (3.269), in order to conclude that

Relation (3.348) together with (3.275) allow us to apply Lemma 3.8 in order to conclude that

• the limit functions ϱ u satisfy the renormalized Eq. (3.325),

• relation (3.328) holds.

Thus we have rigorously justified the strong convergence of {ϱ _δ } _{δ > 0} claimed in (3.329). The proof of Theorem 3.1 is complete.

The reader will have noticed that the weak solutions constructed in the course of the proof of Theorem 3.1 enjoy slightly better regularity and integrability properties than those required in Sect. 2.1. As a matter of fact, the uniform bounds obtained above can be verified for any weak solution of the Navier-Stokes-Fourier system in the sense of Sect. 2.1 and not only for the specific one resulting from our approximation procedure. Similarly, the restrictions on the geometry of the spatial domain can be considerably relaxed and other types of domains, for instance, the periodic slab, can be handled.

■       Regularity of the Weak Solutions:

Theorem 3.2

Let \(\Omega \subset \mathbb{R}^{3}\) be a bounded Lipschitz domain. Assume the data ϱ ₀ , (ϱ u)₀ , E ₀ , (ϱs)₀ , the source terms f , \(\mathcal{Q}\) , the thermodynamic functions p, e, s, and the transport coefficients μ, η, κ satisfy the structural hypotheses ( 3.1 )–( 3.23 ) listed in Sect.  3.1 . Let {ϱ, u, ϑ} be a weak solution to the Navier-Stokes-Fourier system on \((0,T) \times \Omega\) in the sense specified in Sect.  2.1 .

Then, in addition to the minimal integrability and regularity properties required in ( 2.5 )–(2.6), (2.13)–(2.15), (2.30)–(2.31), there holds:

1. (i)
   $$\displaystyle{ \begin{array}{c} \varrho \in C_{\mathrm{weak}}([0,T];L^{\frac{5} {3} }(\Omega )) \cap C([0,T];L^{1}(\Omega )),\\ \\ \varrho \mathbf{u} \in C_{\mathrm{weak}}([0,T];L^{\frac{5} {4} }(\Omega )),\end{array} }$$

   (3.349)
   $$\displaystyle{ \begin{array}{c} \vartheta \in L^{2}(0,T;W^{1,2}(\Omega )) \cap L^{\infty }(0,T;L^{4}(\Omega )),\\ \\ \log \vartheta \in L^{2}(0,T;W^{1,2}(\Omega )),\end{array} }$$

   (3.350)
   $$\displaystyle{ \left \{\begin{array}{c} \mathbb{S}(\vartheta,\nabla _{x}\mathbf{u}) \in L^{q}((0,T) \times \Omega; \mathbb{R}^{3\times 3})\quad \mathit{\mbox{ for a certain }}q> 1,\\ \\ \mathbf{u} \in L^{q}(0,T;W^{1,p}(\Omega; \mathbb{R}^{3}))\ \mathit{\mbox{ for}}\ q = \frac{6} {4-\alpha },p = \frac{18} {10-\alpha },\end{array} \right \} }$$

   (3.351)
   $$\displaystyle{ \left \{\begin{array}{c} \varrho \in L^{q}((0,T) \times \Omega )\quad \mathit{\mbox{ for a certain }}q> \frac{5} {3},\\ \\ p(\varrho,\vartheta ) \in L^{q}((0,T) \times \Omega )\quad \mathit{\mbox{ for a certain }}q> 1.\end{array} \right \} }$$

   (3.352)
2. (ii)

   The total kinetic energy \(\int _{\Omega }\frac{\vert \varrho \mathbf{u}\vert ^{2}} {\varrho } 1_{\{\varrho>0\}}\,\mathrm{d}x\) is lower semicontinuous on (0, T), left lower semicontinuous at T and right lower semicontinuous at 0; in particular

   $$\displaystyle{ \liminf _{t\rightarrow 0+}\Big(\int _{\Omega }\frac{\vert \varrho \mathbf{u}\vert ^{2}} {\varrho } 1_{\{\varrho>0\}}\,\mathrm{d}x\Big)(t) \geq \int _{\Omega }\frac{\vert (\varrho \mathbf{u})_{0}\vert ^{2}} {\varrho _{0}} 1_{\{\varrho _{0}>0\}}\,\mathrm{d}x. }$$

   (3.353)
3. (iii)

   The entropy satisfies

   $$\displaystyle{ \left \{\begin{array}{c} \mathrm{ess}\lim _{t\rightarrow 0+}\int _{\Omega }\varrho s(\varrho,\vartheta )(t,\cdot )\varphi \,\mathrm{d}x\, \geq \int _{\Omega }\varrho _{0}s(\varrho _{0},\vartheta _{0})\varphi \,\mathrm{d}x\\ \\ \mathit{\mbox{ for any}}\ \varphi \in C_{c}^{\infty }(\overline{\Omega }),\ \varphi \geq 0.\end{array} \right \} }$$

   (3.354)

   If, in addition, \(\vartheta _{0} \in W^{1,\infty }(\Omega )\) , then

   $$\displaystyle{ \mathrm{ess}\lim _{t\rightarrow 0+}\int _{\Omega }\varrho s(\varrho,\vartheta )(t,\cdot )\varphi \,\mathrm{d}x =\int _{\Omega }\varrho _{0}s(\varrho _{0},\vartheta _{0})\varphi \,\mathrm{d}x,\ \mathit{\mbox{ for all}}\ \varphi \in C_{c}^{\infty }(\overline{\Omega }). }$$

   (3.355)

Proof

Step 1: Unlike the proof of existence based on the classical theory of parabolic equations requiring \(\Omega\) to be a regular domain, the integrability properties (3.349)–(3.352) of the weak solutions follow directly from the total dissipation balance (2.52) and the space-time pressure estimates obtained by means of the operator \(\mathcal{B}\approx \mathrm{div}_{x}^{-1}\) introduced in Sect. 2.2.5; for more details, see estimates (2.40), (2.46), (2.66), (2.68), (2.73), (2.96) and (2.98). In particular, it is enough to assume \(\Omega \subset \mathbb{R}^{3}\) to be a bounded Lipschitz domain.

Step 2: Strong continuity in time of the density claimed in (3.349) is a general property of the renormalized solutions that follows from the DiPerna and Lions transport theory [85], see Lemma 11.14 in Appendix. Once \(\varrho \in C([0,T];L^{1}(\Omega ))) \cap C_{\mathrm{weak}}([0,T];L^{\frac{5} {3} }(\Omega ))\), we deduce from the momentum equation (2.9) and estimates (3.350)–(3.352) that one may take a representative of \(\mathbf{u} \in L^{q}(0,T;W^{1,p}(\Omega ))\) such that \(\mathbf{m}:=\varrho \mathbf{u} \in C_{\mathrm{weak}}([0,T];L^{\frac{5} {4} }(\Omega; \mathbb{R}^{3}))\). In addition, we may infer from the inequality

that m(t) vanishes almost anywhere on the set \(\{x \in \Omega \,\vert \,\varrho (t) = 0\}\). The expression \(\frac{\vert \mathbf{m}(t)\vert ^{2}} {\varrho (t)} 1_{\{\varrho (t)>0\}}\) is therefore defined for all t ∈ [0, T] and is equal to ϱ | u |²(t) a.a. on (0, T).

Since \(\int _{\Omega }\frac{\vert \mathbf{m}(t)\vert ^{2}} {\varrho (t)+\varepsilon } \,\mathrm{d}x \leq \|\varrho \mathbf{u}\|_{L^{\infty }(0,T;L^{1}(\Omega ))}\) uniformly with ɛ → 0+, we deduce by the Beppo-Lévi monotone convergence theorem that

This information together with (3.349) guarantees \(\mathbf{m}(t)/\sqrt{\varrho (t) +\varepsilon }\) ∈ C _weak([0, T]; \(L^{2}(\Omega ))\). Therefore, for any α > 0 and sufficiently small 0 < ɛ < ɛ(α), and for any τ ∈ [0, T),

where, to justify the inequality in the middle, we have used (3.349) and the lower weak semicontinuity of convex functionals discussed in Theorem 11.27 in Appendix. We have completed the proof of lower semicontinuity in time of the total kinetic energy, and, in particular, formula (3.353).

Step 3: In agreement with formulas (1.11)–(1.12), we deduce from the entropy balance (2.27) that

where the measures [ϱs(ϱ, ϑ)](τ+), τ ∈ [0, T) and [ϱs(ϱ, ϑ)](τ−), τ ∈ (0, T] are defined in the following way

In this formula, \(\zeta \in C(\overline{\Omega })\), I _τ, δ ⁺ = (τ, τ + δ), I _τ, δ ⁻ = (τ −δ, τ) and \(\psi _{\delta }^{(\tau,\pm )} \in C^{1}(\mathbb{R})\) are non increasing functions such that

According to the theorem about the Lebesgue points applied to function ϱs(ϱ, ϑ) (belonging to \(L^{\infty }(0,T;L^{1}(\Omega )\)), we may infer

Letting δ → 0+ in (3.356), we obtain

In the remaining part of the proof, we shall show that

Step 4: To this end we employ in the entropy balance (2.27) the test function φ(t, x) = ψ _δ ^(τ, +)(t)ϑ ₀(x), τ ∈ (0, T), which is admissible provided \(\vartheta _{0} \in W^{1,\infty }(\Omega )\). Using additionally (3.357), we get

for a.a. τ ∈ (0, T). On the other hand, the total energy balance (2.22) with the test function ψ = ψ _δ ^(τ, +) yields

for a.a. τ ∈ (0, T). Now, we introduce the Helmholtz function

and combine (3.360)–(3.361) to get

for a.a. τ in (0, T).

It follows from the thermodynamic stability hypothesis (1.44) that \(\varrho \mapsto H_{\vartheta _{0}}(\varrho,\vartheta _{0})\) is strictly convex for any fixed ϑ ₀ and that \(\vartheta \mapsto H_{\vartheta _{0}}(\varrho,\vartheta )\) attains its global minimum at ϑ ₀, see Sect. 2.2.3 for more details. Consequently,

Moreover, due to the strong continuity of density with respect to time stated in (3.349), we show

while the last integral at the right hand side of (3.362) tends to 0 as τ → 0+ since the integrand belongs to \(L^{1}((0,T) \times \Omega )\). Thus, relation (3.362) reduces in the limit τ → 0+ to

whence \(\mathrm{ess}\,\mathrm{lim}_{\tau \rightarrow 0+}\sigma \left [[0,\tau ] \times \overline{\Omega }\right ] = 0\) and (3.359) follows. Having in mind identity (3.357), statement (3.355) now follows by letting τ → 0+ in (3.358) (evidently, the right hand side in (3.358) tends to zero as the integrand belongs to \(L^{1}((0,T) \times \Omega )\)).

Theorem 3.2 is proved.

□