On the high friction limit for the complete Euler system

Abstract

We show that solutions of the complete Euler system of gas dynamics perturbed by a friction term converge to a solution of the porous medium equation in the high friction/long time limit. The result holds in the largest possible class of generalized solutions–the measure–valued solutions of the Euler system.

1 Introduction

In a recent paper, Lattanzio and Tzavaras [13] consider the singular limit of the isentropic Euler system perturbed by a high friction term. Our goal is to extend this result to the physically more adequate setting of the complete Euler system of gas dynamics, where temperature changes as well as possible singularities resulting in the increase of the total entropy of the system are allowed.

1.1 Euler system of gas dynamics

We consider a scaled Euler system of gas dynamics in the form:

Equation of continuity:

$$\begin{aligned} \varepsilon \partial _t \varrho + {\textrm{div}}_{x}\textbf{m}= 0. \end{aligned}$$

(1.1)

Momentum equation:

$$\begin{aligned} \varepsilon \partial _t \textbf{m}+ {\textrm{div}}_{x}\left( \frac{\textbf{m}\otimes \textbf{m}}{\varrho } \right) + \nabla _{x}p = - \frac{1}{\varepsilon } \textbf{m}. \end{aligned}$$

(1.2)

Energy balance:

$$\begin{aligned} \varepsilon \partial _t \left( \frac{1}{2} \frac{|\textbf{m}|^2}{\varrho } + \varrho e \right) + {\textrm{div}}_{x}\left[ \left( \frac{1}{2} \frac{|\textbf{m}|^2}{\varrho } + \varrho e + p \right) \frac{\textbf{m}}{\varrho } \right] = - \frac{1}{\varepsilon } \frac{|\textbf{m}|^2}{\varrho }. \end{aligned}$$

(1.3)

The pressure p and the internal energy e are thermodynamic functions interrelated through Gibbs’ equation

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

(1.4)

where $\vartheta$ is the (absolute) temperature and s the (specific) entropy. Here, D refers to differential with respect to the independent thermodynamics variables, e.g. $\varrho , \vartheta$. In the present paper, rather general constitutive relations are supposed, with only minor restrictions specified in Theorem 2.2 below. We consider the fluid mass density $\varrho = \varrho (t,x)$, the momentum $\textbf{m}= \textbf{m}(t,x)$, together with the total entropy $S = (\varrho s)(t,x)$ as the basic state variables (unknowns) in the system of equations (1.1)–(1.3). The fluid is confined to a bounded domains $\Omega \subset R^d$ and the problem is formally closed by imposing the impermeability boundary conditions

$$\begin{aligned} \textbf{m}\cdot \textbf{n}|_{\partial \Omega } = 0, \end{aligned}$$

(1.5)

and the initial conditions

$$\begin{aligned} \varrho (0, \cdot ) = \varrho _0,\ \textbf{m}(0, \cdot ) = \textbf{m}_0,\ S(0, \cdot ) = S_0. \end{aligned}$$

(1.6)

1.2 Large friction limit

Smooth solutions of the Euler system (1.1)–(1.3) conserve entropy, specifically, it follows from Gibbs’ relation (1.4)

$$\begin{aligned} \varepsilon \partial _t S + {\textrm{div}}_{x}\left( S \frac{\textbf{m}}{\varrho } \right) = 0. \end{aligned}$$

(1.7)

In particular, if $s_0 = s(0, \cdot ) = {\overline{s}}$ is constant, it follows from (1.7) $s(t, \cdot ) = {\overline{s}}$ as long as the solution remains smooth and (1.1)–(1.3) reduces to the so-called isentropic Euler system for only two unknowns $\varrho$ and $\textbf{m}$.

The term $\frac{1}{\varepsilon } \textbf{m}$ in the momentum equation, together with its counterpart $\frac{1}{\varepsilon } \frac{|\textbf{m}|^2}{\varrho }$ in the energy balance, represent the effect of “friction” on the gas motion. High friction regularization of the simplified Euler system has been studied by several authors, see Dafermos and Pan [6], Sideris, Thomases, and Wang [15], and the references cited therein.

Lattanzio and Tzavaras [13] identified the high friction limit $\varepsilon \rightarrow 0$ in the isentropic Euler system as the porous medium equation:

$$\begin{aligned} \varrho _{\varepsilon }\rightarrow r,\ \text{ where }\ \partial _t r - \Delta _{x}p(r, r {\overline{s}}) = 0,\ {\overline{s}} - \text{ constant. } \end{aligned}$$

(1.8)

The proof in [13] is formal, conditioned by the existence of weak solutions to the scaled isentropic Euler system with the life span independent of the scaling parameter $\varepsilon$. As a matter of fact, this type of relaxation limits has been studied before in different contexts and with the use of different techniques, see e.g. Marcati and Milani [14]. The reference [13] presents a systematic study of this kind of relaxation limits using the relative energy techniques, again, see also Lattanzio and Tzavaras [12], where the relative energy is also used in the context of measure-valued solutions.

The motion of a gas obeying the physically relevant complete Euler system (1.1)–(1.3) is not likely to be isentropic, with possible shocks developed in a finite time violating the entropy equation (1.7). A suitable admissibility condition for the general weak solutions is then formulated in terms of the entropy inequality

$$\begin{aligned} \varepsilon \partial _t S + {\textrm{div}}_{x}\left( S \frac{\textbf{m}}{\varrho } \right) \ge 0, \end{aligned}$$

or its renormalized variant (see e.g. Chen and Frid [5])

$$\begin{aligned} \varepsilon \partial _t \left( \varrho \chi \left( \frac{S}{\varrho } \right) \right) + {\textrm{div}}_{x}\left[ \chi \left( \frac{S}{\varrho } \right) \textbf{m}\right] \ge 0 \end{aligned}$$

(1.9)

for any $\chi$ concave, $\chi ' \ge 0$, $\chi$ bounded above.

Our main goal is to extend the result of Lattanzio and Tzavaras [13] to the complete Euler system. The limit density profile r and the entropy $s = \frac{S}{r}$ (formally) satisfy the following system of equations:

$$\begin{aligned} \partial _t r - \Delta _{x}p(r, rs)&= 0, \end{aligned}$$

(1.10)

$$\begin{aligned} \partial _t s - \frac{1}{r} \nabla _{x}p (r, rs) \cdot \nabla _{x}s&= 0, \end{aligned}$$

(1.11)

supplemented with the initial and boundary data

$$\begin{aligned} r(0, \cdot ) = r_0,\ s(0, \cdot ) = s_0,\ \nabla _{x}p(r, rs) \cdot \textbf{n}|_{\partial \Omega } = 0. \end{aligned}$$

(1.12)

Accordingly, we suppose that the initial data of the Euler system converge to the initial conditions of (1.10), (1.11) specifically,

$$\begin{aligned} \varrho _0 = \varrho _{0,\varepsilon } \rightarrow r_0,\ S_0 = S_{0, \varepsilon } \rightarrow r_0 s_0, \ \text{ and }\ \textbf{m}_0 = \textbf{m}_{0,\varepsilon } \rightarrow 0. \end{aligned}$$

(1.13)

Solvability of the problem (1.10), (1.11) is discussed in the forthcoming section.

Although the recent results based on the method of convex integration provide weak solutions (even infinitely many) for a large class of initial data, see e.g. [8], a general existence result that would cover all finite energy initial data is not available so far. For this reason, we consider a larger class of measure–valued solutions in the spirit of [3]. The advantage of considering the measure–valued solutions of the Euler system is not only their existence that may be shown for all physically admissible data, see e.g. Kröner and Zaja̧czkowski [11] or the more recent adaptation of the same method of construction in [1]. The measure–valued solutions capture a large variety of singular limits including the low diffusion limit of the Navier–Stokes–Fourier system [2] as well as the alternative model proposed by Brenner, see [9, Chapter 10]. The truly measure–valued solutions, unlike the Euler system, can also mimick the behaviour of complete fluid systems in highly turbulent regime, see [7].

The paper is organized as follows. In Sect. 2, we introduce the concept of measure–valued solution and state the main result concerning the high friction limit. In Sect. 3, we recall the relative energy inequality introduced in [3] and further developed in [9] and prove the desired convergence.

2 Measure–valued solutions, main result

Besides the general Gibbs’ relation (1.4), we suppose that the thermodynamic functions p, e, and s satisfy the hypothesis of thermodynamic stability. This can be conveniently formulated in terms of the variables $(\varrho , S)$ as convexity of the internal energy:

$$\begin{aligned} E_{\textrm{int}}: (\varrho , S)&\in R^{2} \mapsto \varrho e(\varrho , S) \in [0, \infty ] \ \text{ is } \text{ a } \text{ convex } \text{ l.s.c } \text{ function }, \nonumber \\ E_{\textrm{int}}&\in C^2 \left( \textrm{int}( \textrm{dom} ) [E_{\textrm{int}}] \right) ,\ \nabla ^2 E_{\textrm{int}} > 0, \end{aligned}$$

(2.1)

cf. [9, Chapter 4, Section 4.1.6]. Note that for the standard Boyle-Mariotte equation of state

$$\begin{aligned} p = (\gamma - 1) \varrho e,\ e = c_v \vartheta ,\ c_v = \frac{1}{\gamma - 1}, \ \gamma > 1, \end{aligned}$$

we have

$$\begin{aligned} p(\varrho , S) = (\gamma - 1) E_{\textrm{int}}(\varrho ,S) = \left\{ \begin{array}{ll} \varrho ^\gamma \exp \left( \frac{S}{c_v \varrho } \right) &{} \text{ if }\ \varrho > 0, \nonumber \\ 0 &{} \text{ if }\ \varrho = 0,\ S \le 0, \nonumber \\ \infty &{} \text{ otherwise, } \end{array}\right. \nonumber \end{aligned}$$

see [9, Chapter 2, Section 2.2.4].

Similarly, we define the kinetic energy,

$$\begin{aligned} E_{\textrm{kin}}: (\varrho , \textbf{m}) \mapsto \left\{ \begin{array}{ll} \frac{1}{2} \frac{ |\textbf{m}|^2 }{\varrho } &{} \text{ if } \ \varrho > 0, \\ 0 &{}\text{ if }\ \varrho = 0, \ \textbf{m}= 0, \\ \infty &{} \text{ otherwise, } \end{array}\right. \end{aligned}$$

and the total energy

$$\begin{aligned} E(\varrho , S, \textbf{m})&= E_{\textrm{kin}}(\varrho , \textbf{m}) + E_{\textrm{int}}(\varrho , S) \\&\text{ a } \text{ convex } \text{ l.s.c. } \text{ function } \text{ on }\ R^{2 + d},\ \ \text{ strictly } \text{ convex } \text{ on } \text{ its } \text{ domain. } \end{aligned}$$

2.1 Measure–valued solutions

Following [3], we define measure–valued solution of the Euler system (1.1)–(1.3), (1.9), (1.5)–(1.6) a weakly measurable family of Borel probability measures,

$$\begin{aligned} {\mathcal {V}}: (t, x) \in (0,T) \times \Omega \mapsto {\mathcal {V}}_{t,x} \in {\mathfrak {P}} [R \times R \times R^d]. \end{aligned}$$

Moreover, we denote

$$\begin{aligned} \left\langle {\mathcal {V}}_{t,x}; F({\widetilde{\varrho }}, {\widetilde{S}}, {\widetilde{\textbf{m}}}) \right\rangle = \int _{R^{2+d}} F({\widetilde{\varrho }}, {\widetilde{S}}, {\widetilde{\textbf{m}}}) \ \textrm{d}{\mathcal {V}}_{t,x}, \end{aligned}$$

for any Borel measurable function F of the “dummy” variables $({\widetilde{\varrho }}, {\widetilde{S}}, {\widetilde{\textbf{m}}}) \in R^{2 + d}$.

Definition 2.1

(Measure–valued solution of the Euler system)

We say that a parametrized family $( {\mathcal {V}}_{t,x} )_{(t,x) \in (0,T) \times \Omega }$ of probability measures on the Euclidean space $R^{2 + d}$ is a measure–valued solution to the Euler system (1.1)–(1.3), with the boundary conditions (1.5), and the initial data (1.6), if the following holds:

• Compatibility:

  $$\begin{aligned} {\mathcal {V}}_{t,x} \left\{ {\widetilde{\varrho }}\ge 0 \right\} = {\mathcal {V}}_{t,x} \left\{ {\widetilde{\varrho }}= 0,\ {\widetilde{S}}\ge 0 \right\} = 1 \ \text{ for } \text{ a.a. }\ (t,x) \in (0,T) \times \Omega .\nonumber \\ \end{aligned}$$

  (2.2)

• Equation of continuity:

  $$\begin{aligned} \int _0^T \int _{\Omega } \Big ( \varepsilon \left\langle {\mathcal {V}}_{t,x}; {\widetilde{\varrho }}\right\rangle \partial _t \varphi (t,x) + \left\langle {\mathcal {V}}_{t,x}; {\widetilde{\textbf{m}}}\right\rangle \cdot \nabla _{x}\varphi (t,x) \Big ) \ \,\textrm{d} {x} \,\textrm{d} t = - \varepsilon \int _{\Omega } \varrho _0 \varphi (0, x) \ \,\textrm{d} {x}; \nonumber \\ \end{aligned}$$

  (2.3)

  for any $\varphi \in C^1_c([0,T) \times {\overline{\Omega }})$.

• Momentum equation: There exists a tensor–valued measure

  $$\begin{aligned} {\mathfrak {R}} \in L^{\infty }_{\textrm{weak}}(0,T; {\mathcal {M}}_{\textrm{sym}}^+({\overline{\Omega }}; R^{d \times d})) \end{aligned}$$

  such that the integral identity

  $$\begin{aligned}&\int _0^T \int _{\Omega } \left( \varepsilon \left\langle {\mathcal {V}}_{t,x}; {\widetilde{\textbf{m}}}\right\rangle \partial _t {\varvec{\varphi }}(t,x) + \left\langle {\mathcal {V}}_{t,x}; \mathbbm {1}_{{\widetilde{\varrho }}> 0} \frac{{\widetilde{\textbf{m}}}\otimes {\widetilde{\textbf{m}}}}{{\widetilde{\varrho }}} \right\rangle : \nabla _{x}{\varvec{\varphi }}(t,x) \right) \ \,\textrm{d} {x} \,\textrm{d} t \nonumber \\&\qquad + \int _0^T \int _{\Omega } \left\langle {\mathcal {V}}_{t,x}; p({\widetilde{\varrho }}, {\widetilde{S}}) \right\rangle {\textrm{div}}_{x}{\varvec{\varphi }}(t,x) \ \,\textrm{d} {x} \,\textrm{d} t \nonumber \\&\quad = \int _0^T \int _{\Omega } \frac{1}{\varepsilon } \left\langle {\mathcal {V}}_{t,x}; {\widetilde{\textbf{m}}}\right\rangle \cdot {\varvec{\varphi }}(t,x) \ \,\textrm{d} {x} \,\textrm{d} t - \int _0^T \int _{{\overline{\Omega }}} \nabla _{x}{\varvec{\varphi }}(t,x) : \textrm{d}{\mathfrak {R}} \,\textrm{d} t \nonumber \\&\qquad - \varepsilon \int _{\Omega } \textbf{m}_0(x) \cdot {\varvec{\varphi }}(0, x) \ \,\textrm{d} {x}; \end{aligned}$$

  (2.4)

  holds for any ${\varvec{\varphi }}\in C^1_c([0,T) \times {\overline{\Omega }}; R^d)$, ${\varvec{\varphi }}\cdot \textbf{n}|_{\partial \Omega } = 0$.

• Entropy inequality:

  $$\begin{aligned} \int _0^T&\int _{\Omega } \left( \varepsilon \left\langle {\mathcal {V}}_{t,x}; {\widetilde{\varrho }}\chi \left( \frac{{\widetilde{S}}}{{\widetilde{\varrho }}} \right) \right\rangle \partial _t \varphi (t,x) + \left\langle {\mathcal {V}}_{t,x}; {\widetilde{\varrho }}\chi \left( \frac{{\widetilde{S}}}{{\widetilde{\varrho }}} \right) \frac{{\widetilde{\textbf{m}}}}{{\widetilde{\varrho }}} \right\rangle \cdot \nabla _{x}\varphi (t,x) \right) \ \,\textrm{d} {x} \,\textrm{d} t \nonumber \\&\le - \varepsilon \int _{\Omega } \varrho _0(x) \chi \left( \frac{S_0 (x) }{\varrho _0(x) } \right) \varphi (0, x) \ \,\textrm{d} {x} \end{aligned}$$

  (2.5)

  for any $\varphi \in C^1_c([0,T) \times {\overline{\Omega }})$, $\varphi \ge 0$, and any $\chi \in C(R)$ concave, $\chi ' \ge 0$, $\chi \le {\overline{\chi }} \in R$.

• Energy inequality: There exists a constant $\Lambda > 0$ such that

  $$\begin{aligned}&\int _{\Omega } \left\langle {\mathcal {V}}_{\tau ,x} ; E({\widetilde{\varrho }}, {\widetilde{S}}, {\widetilde{\textbf{m}}}) \right\rangle \ \,\textrm{d} {x} + \Lambda \int _{{\overline{\Omega }}} \textrm{d}\ \textrm{trace}[{\mathfrak {R}} (\tau , \cdot ) ] + \frac{1}{\varepsilon ^2} \int _0^\tau \int _{\Omega } \left\langle {\mathcal {V}}_{t,x} ; \frac{|{\widetilde{\textbf{m}}}|^2}{{\widetilde{\varrho }}} \right\rangle \ \,\textrm{d} {x} \,\textrm{d} t \nonumber \\&\quad \le \int _{\Omega } E(\varrho _0, S_0, \textbf{m}_0) \ \,\textrm{d} {x} \end{aligned}$$

  (2.6)

  for a.a. $\tau \in (0,T)$.

The measure valued solutions were introduced in [3] to capture the largest class possible of limits of the so–called consistent approximations of the Euler system. Their existence and even possibility to construct a solution semigroup was shown in [1, Proposition 3.8]. Besides, the numerical approximations introduced in [9, Chapter 10] are examples of consistent approximations and the convergence results presented therein may be seen as another way of identifying measure valued solutions.

2.2 Solvability of the limit problem

We point out that the hypothesis of thermodynamic stability enforced through strict convexity of $E_{\textrm{int}} = E_{\textrm{int}}(\varrho , S)$ has important consequence on solvability of the system (1.10), (1.11) at least in the case of constant initial entropy $s_0 = {\overline{s}}$. Indeed the entropy balance (1.11) implies $s (t,x) = {\overline{s}}$ for any t, x independently of r. Accordingly, the density profile r can be determined as a solution of the problem

$$\begin{aligned} \partial _t r - {\textrm{div}}_{x}\left[ \Big ( \partial _{\varrho } p(r, r {\overline{s}}) + {\overline{s}} \partial _S p (r, r {\overline{s}}) \Big ) \nabla _{x}r \right] , \ r(0, \cdot ) = r_0,\ \nabla _{x}r \cdot \textbf{n}|_{\partial \Omega } = 0.\nonumber \\ \end{aligned}$$

(2.7)

It follows form Gibbs’ Eq. (1.4) that

$$\begin{aligned} p(\varrho , S) = \frac{ \partial (\varrho e(\varrho ,S))}{\partial \varrho } \varrho + \frac{ \partial (\varrho e(\varrho ,S))}{\partial S}S - \varrho e(\varrho ,S). \end{aligned}$$

Consequently,

$$\begin{aligned} \frac{ \partial p(\varrho ,S)}{\partial \varrho } = \frac{\partial ^2 (\varrho e (\varrho ,S))}{\partial ^2 \varrho } \varrho + \frac{\partial ^2 (\varrho e(\varrho ,S))}{\partial \varrho \partial S} S, \end{aligned}$$

and, similarly,

$$\begin{aligned} \frac{ \partial p(\varrho ,S)}{\partial S} = \frac{\partial ^2 (\varrho e (\varrho ,S))}{\partial \varrho \partial S} \varrho + \frac{\partial ^2 (\varrho e(\varrho ,S))}{\partial ^2 S} S. \end{aligned}$$

Thus we compute

$$\begin{aligned} \partial _{\varrho } p(r, r {\overline{s}}) + {\overline{s}} \partial _S p (r, r {\overline{s}}) = r \left( \frac{\partial ^2 (r e (r,r {\overline{s}}))}{\partial ^2 \varrho } + 2 \frac{\partial ^2 (r e(r,r {\overline{s}} ))}{\partial \varrho \partial S} {\overline{s}} + \frac{\partial ^2 (r e(r,r {\overline{s}} ))}{\partial ^2 S} |{\overline{s}}|^2 \right) . \end{aligned}$$

(2.8)

Since the internal energy $E_{\textrm{int}}$ is a strictly convex function of $(\varrho , S)$, we get

$$\begin{aligned} \frac{\partial ^2 (r e (r,r {\overline{s}}))}{\partial ^2 \varrho }> 0,\ \frac{\partial ^2 (r e(r,r {\overline{s}} ))}{\partial ^2 S} > 0, \end{aligned}$$

and

$$\begin{aligned} \frac{\partial ^2 (r e (r,r {\overline{s}}))}{\partial ^2 \varrho }\frac{\partial ^2 (r e(r,r {\overline{s}} ))}{\partial ^2 S} > \left| \frac{\partial ^2 (r e(r,r {\overline{s}} ))}{\partial \varrho \partial S} \right| ^2. \end{aligned}$$

This yields

$$\begin{aligned} \Big ( \partial _{\varrho } p(r, r {\overline{s}}) + {\overline{s}} \partial _S p (r, r {\overline{s}}) \Big )> 0 \ \text{ whenever } \ r > 0, \end{aligned}$$

(2.9)

and, consequently, (2.7) is a non–degenerate parabolic equation as soon as $r_0 > 0$.

2.3 Main result, high friction limit

To begin, we introduce the relative energy expressed in the variables $(\varrho , S, \textbf{m})$,

$$\begin{aligned}&E \left( \varrho , S, \textbf{m}\Big | r, {\widehat{S}}, r \textbf{U}\right) \nonumber \\&\quad = \varrho \left| \frac{\textbf{m}}{\varrho } - \textbf{U}\right| ^2 + \varrho e(\varrho , S) - \frac{ \partial (\varrho e(\varrho ,S)) }{\partial \varrho }(r, {\widehat{S}})(\varrho - r) - \frac{ \partial (\varrho e(\varrho ,S)) }{\partial S}(r, {\widehat{S}})(S - {\widehat{S}}) \nonumber \\&\qquad - r e(r, {\widehat{S}}). \end{aligned}$$

(2.10)

More precisely, the relative energy for fixed values of the parameters $(r, {\widehat{S}}, \textbf{U})$ is a convex l.s.c. function of $(\varrho , S, \textbf{m})$ defined as

$$\begin{aligned}{} & {} E \left( \varrho , S, \textbf{m}\Big | r, {\widehat{S}}, r \textbf{U}\right) = E (\varrho , S, \textbf{m}) - \partial _{\varrho , S, \textbf{m}} E (r, {{\widehat{S}}}, r \textbf{U}) \cdot (\varrho - r, S - {\widehat{S}}, \textbf{m}- r \textbf{U}) \\{} & {} \quad - E(r, {\widehat{S}}, r \textbf{U}). \end{aligned}$$

In particular, if $(r, {\widetilde{S}}, r \textbf{U}) \in \textrm{int}(\textrm{dom}(E))$, then

$$\begin{aligned} E \left( \varrho , S, \textbf{m}\Big | r, {\widehat{S}}, r \textbf{U}\right) = 0 \ \Leftrightarrow \ \varrho = r,\ S = {\widehat{S}},\ \textbf{m}= r \textbf{U}, \end{aligned}$$

see [9, Chapter 4, Section 4.1.6] for details.

We are ready to state our main result concerning the high friction limit.

Theorem 2.2

(High friction asymptotic limit)

Let $\Omega \subset R^d$ be a bounded domain of class $C^{2 + \nu }$. In addition to Gibbs’ relation (1.4) and the hypothesis of thermodynamics stability (2.1), let the thermodynamic functions p, e satisfy

$$\begin{aligned} p \le {\overline{p}} (1 + \varrho e ) \ \text{ for } \text{ some } \text{ constant }\ {\overline{p}}. \end{aligned}$$

(2.11)

Finally, suppose that the initial data satisfy

$$\begin{aligned} \varrho _{0, \varepsilon } > 0,\ S_{0,\varepsilon }&\ge \varrho _{0,\varepsilon } {\underline{s}} \ \quad \text{ for } \text{ some }\ {\underline{s}} \in R, \nonumber \\ \int _{\Omega } E \left( \varrho _{0, \varepsilon }, S_{0, \varepsilon }, \textbf{m}_{0, \varepsilon }\ \Big | \ r_0, r_0 s_0, 0 \right) \ \,\textrm{d} {x}&\rightarrow 0 \ \text{ as }\ \varepsilon \rightarrow 0, \end{aligned}$$

(2.12)

where

$$\begin{aligned} r_0, s_0 \in C^2({\overline{\Omega }}),\ (r_0, r_0 s_0 ) \in \textrm{int}( \textrm{dom} E_{\textrm{int}} ). \end{aligned}$$

(2.13)

Let $({\mathcal {V}}^\varepsilon _{t,x})_{\varepsilon > 0}$ be a family of measure–valued solutions of the Euler system with the initial data $(\varrho _{0,\varepsilon }, S_{0, \varepsilon }, \textbf{m}_{0,\varepsilon } )_{\varepsilon > 0}$ in the sense of Definition 2.1. Suppose that the limit system (1.10), (1.11), (1.12) admits a $C^2-$solution r, s such that

$$\begin{aligned} (r, r s ) \in {\textrm{int}( \textrm{dom}) [E_{\textrm{int}} ]}\ \text{ for } \text{ any }\ t\in [0,T]. \end{aligned}$$

(2.14)

Then

$$\begin{aligned} \textrm{ess} \sup _{\tau \in (0,T)} \int _{\Omega } \left\langle {\mathcal {V}}^\varepsilon _{\tau , x}; E \left( {\widetilde{\varrho }}, {\widetilde{S}}, {\widetilde{\textbf{m}}}\ \Big | \ r(\tau ,x), r s (\tau ,x), 0 \right) \right\rangle \ \,\textrm{d} {x} \rightarrow 0 \ \text{ as } \ \varepsilon \rightarrow 0.\nonumber \\ \end{aligned}$$

(2.15)

The remaining part of the paper is devoted to the proof of Theorem 2.2.

Remark 2.3

Noticing that the total energy is stricly convex, meaning the relative energy represents a Bregman distance, we can reformulate the convergence statement (2.15) in terms of probability theory:

$$\begin{aligned} \textrm{ess} \sup _{\tau \in (0,T)} \int _{\Omega } W_1 \left[ {\mathcal {V}}^\varepsilon _{\tau ,x}; \delta _{( r(\tau ,x), rs (\tau ,x), 0 )} \right] \ \,\textrm{d} {x} \rightarrow 0 \ \text{ as }\ \varepsilon \rightarrow 0, \end{aligned}$$

where $W_1$ denotes the Monge–Kantorowich (Wasserstein – 1) distance between probability measures, and $\delta _X$ stands for the Dirac mass at X.

Remark 2.4

Solvability of the limit problem (1.10), (1.11), (1.12) has been discussed in Sect. 2.2 on condition that the initial entropy $s_0$ is constant. Note that in this case (2.14) follows from (2.13) by the maximum principle. Another, a rather trivial situation when the limit problem is solvable globally in time, is the choice of the initial data

$$\begin{aligned} p(r_0, r_0 s_0) = {\overline{p}} \ \text{-- } \text{ a } \text{ positive } \text{ constant. } \end{aligned}$$

Indeed $r = r_0$, $s = s_0$ then obviously solve the problem. Unlike in Sect. 2.2, the initial entropy distribution may effectively depend on x.

Remark 2.5

The Proof of Theorem 2.2 presented below will actually yield a more exact rate of convergence (2.15):

$$\begin{aligned} \textrm{ess} \sup _{\tau \in (0,T)} \int _{\Omega } \left\langle {\mathcal {V}}^\varepsilon _{\tau , x}; E \left( {\widetilde{\varrho }}, {\widetilde{S}}, {\widetilde{\textbf{m}}}\ \Big | \ r(\tau ,x) , r s (\tau ,x) , 0 \right) \right\rangle \ \,\textrm{d} {x} \nonumber \\ \le \int _{\Omega } E \left( \varrho _{0, \varepsilon }, S_{0, \varepsilon }, \textbf{m}_{0, \varepsilon }\ \Big | \ r_0, r_0 s_0, 0 \right) \ \,\textrm{d} {x} + c \varepsilon . \end{aligned}$$

(2.16)

3 High friction limit, proof of the main result

Similarly to Lattanzio and Tzavaras [13], the Proof of Theorem 2.2 leans on stability on the limit solution expressed by means of the relative energy inequality. The relevant version of the latter for the complete Euler system was introduced in [3]. The relative energy inequality in the context of measure-valued solutions for isentropic Euler with friction and also nonlocal terms was studied in [4]. Note that the current studies on complete Euler system do not fall into the general framework of hyperbolic systems studied in [10].

3.1 Entropy minimum principle

We start by exploiting boundedness of the initial entropy, namely

$$\begin{aligned} S_{0,\varepsilon } \ge \varrho _{0,\varepsilon } {\underline{s}}, \end{aligned}$$

yielding the entropy minimum principle in the form

$$\begin{aligned} {\mathcal {V}}^{\varepsilon }_{t,x} \left\{ {\widetilde{S}}\ge {\widetilde{\varrho }}{\underline{s}} \ \Big | \ {\widetilde{\varrho }}> 0 \right\} = 1. \end{aligned}$$

(3.1)

Indeed we may choose $\chi$ in the renormalized entropy inequality (2.5) as

$$\begin{aligned} \chi (Z) < 0 \ \text{ for }\ Z \le {\underline{s}},\ \chi (Z) = 0 \ \text{ for }\ Z \ge {\underline{s}}. \end{aligned}$$

Considering spatially homogeneous test function $\varphi$ in (2.5) we deduce

$$\begin{aligned} \int _{\Omega } \left\langle {\mathcal {V}}^\varepsilon _{\tau ,x}; {\widetilde{\varrho }}\chi \left( \frac{{\widetilde{S}}}{{\widetilde{\varrho }}} \right) \right\rangle \ \,\textrm{d} {x} \ge 0 \ \text{ for } \text{ a.a. }\ \tau \ge 0, \end{aligned}$$

which yields the desired conclusion.

Finally, relation (3.1) together with the compatibility condition (2.2) converts (3.1) to an unconditional results

$$\begin{aligned} {\mathcal {V}}^\varepsilon _{t,x} \left\{ {\widetilde{S}}\ge {\widetilde{\varrho }}{\underline{s}} \right\} = 1. \end{aligned}$$

(3.2)

3.2 Relative energy inequality

Recalling the definition (2.10) of the relative energy, let us introduce the temperature $\Theta$ evaluated by means of the variables r and ${\widehat{S}}$ through the implicit relation

$$\begin{aligned} {\widehat{S}} = r s(r, \Theta ). \end{aligned}$$

(3.3)

Under the condition (3.2), the relative energy inequality associated to the Euler system reads, see [3] or [9, Chapter 4, Section 4.1.7, Chapter 6, Section 6.2]:

$$\begin{aligned}&\int _{\Omega } \left\langle {\mathcal {V}}_{\tau ,x} ; E \left( {\widetilde{\varrho }}, {\widetilde{S}}, {\widetilde{\textbf{m}}}\ \Big |\ r, {\widehat{S}} , r \textbf{U}\right) (\tau ,x) \right\rangle \ \,\textrm{d} {x} + \Lambda \int _{\Omega } \textrm{d}\ \textrm{trace} [ {\mathfrak {R}}](\tau , \cdot ) \nonumber \\&\qquad + \frac{1}{\varepsilon ^2} \int _0^\tau \int _{\Omega } \left\langle {\mathcal {V}}_{t,x} ;{\widetilde{\varrho }}\left| \frac{{\widetilde{\textbf{m}}}}{{\widetilde{\varrho }}} - \textbf{U}(t,x) \right| ^2 \right\rangle \ \,\textrm{d} {x} \,\textrm{d} t \nonumber \\&\quad \le \int _{\Omega } E \left( \varrho _0, S_0 , \textbf{m}_0 \ \Big |\ r, {\widehat{S}} , r \textbf{U}\right) (0, x) \ \,\textrm{d} {x} \nonumber \\&\qquad - \frac{1}{\varepsilon } \int _0^\tau \int _{\Omega } \left\langle {\mathcal {V}}_{t,x} ;\mathbbm {1}_{{\widetilde{\varrho }}> 0} \frac{ ({\widetilde{\varrho }}\textbf{U}(t,x) - {\widetilde{\textbf{m}}}) \otimes ({\widetilde{\varrho }}\textbf{U}(t,x) - {\widetilde{\textbf{m}}}) }{{\widetilde{\varrho }}} : \nabla _{x}\textbf{U}(t,x) \right\rangle \ \,\textrm{d} {x} \,\textrm{d} t \nonumber \\&\qquad - \frac{1}{\varepsilon } \int _0^\tau \int _{\Omega } \left\langle {\mathcal {V}}_{t,x} ; \left( p({\widetilde{\varrho }},{\widetilde{S}}) - ({\widetilde{\varrho }}- r) \partial _\varrho p(r, {\widehat{S}}) - ({\widetilde{S}}- {\widehat{S}}) \partial _S p(r,{\widehat{S}} ) \right. \right. \\&\qquad \left. \left. - p(r, {\widehat{S}}) \right) (t,x) \right\rangle {\textrm{div}}_{x}\textbf{U}(t,x) \ \,\textrm{d} {x} \,\textrm{d} t \nonumber \\&\qquad + \int _0^\tau \int _{{\mathbb {T}}^{d}} \left\langle {\mathcal {V}}_{t,x} ; {\widetilde{\varrho }}\textbf{U}(t,x) - {\widetilde{\textbf{m}}}\right\rangle \Big ( \partial _t \textbf{U}+ \frac{1}{\varepsilon } \textbf{U}\cdot \nabla _{x}\textbf{U}+ \frac{1}{\varepsilon r} \nabla _{x}p(r, {\widehat{S}} ) \Big )(t,x) \ \,\textrm{d} {x} \,\textrm{d} t \nonumber \\&\qquad + \int _0^\tau \int _{\Omega } \left\langle {\mathcal {V}}_{t,x} ; r(t,x) - {\widetilde{\varrho }}\right\rangle \frac{1}{r} \partial _\varrho p(r, {\widehat{S}}) \left( \partial _t r + \frac{1}{\varepsilon } {\textrm{div}}_{x}(r \textbf{U}) \right) (t,x) \ \,\textrm{d} {x} \,\textrm{d} t \nonumber \\&\qquad + \int _0^\tau \int _{\Omega } \left\langle {\mathcal {V}}_{t,x} ; r(t,x) - {\widetilde{\varrho }}\right\rangle \frac{1}{r} \partial _S p(r, {\widehat{S}}) \left( \partial _t {\widehat{S}} + \frac{1}{\varepsilon } {\textrm{div}}_{x}( {\widehat{S}} \textbf{U}) \right) (t,x) \ \,\textrm{d} {x} \,\textrm{d} t \nonumber \\&\qquad + \int _0^\tau \int _{\Omega } \left\langle {\mathcal {V}}_{t,x}; \frac{{\widetilde{\varrho }}}{r} {\widehat{S}}(t,x) - {\widetilde{\varrho }}\chi \left( \frac{{\widetilde{S}}}{{\widetilde{\varrho }}} \right) \right\rangle \nonumber \\&\qquad \Big ( \partial _t \Theta + \frac{1}{\varepsilon } \textbf{U}\cdot \nabla _{x}\Theta + \frac{1}{\varepsilon } \partial _S p(r, {\widehat{S}}) {\textrm{div}}_{x}\textbf{U}\Big )(t,x) \ \,\textrm{d} {x} \,\textrm{d} t \nonumber \\&\qquad + \frac{1}{\varepsilon } \int _0^\tau \int _{\Omega } \left\langle {\mathcal {V}}_{t,x} ; \left( {\widetilde{\varrho }}\frac{{\widehat{S}}}{r}(t,x) - {\widetilde{\varrho }}\chi \left( \frac{{\widetilde{S}}}{{\widetilde{\varrho }}} \right) \right) \left( \frac{{\widetilde{\textbf{m}}}}{{\widetilde{\varrho }}} - \textbf{U}(t,x) \right) \right\rangle \cdot \nabla _{x}\Theta (t,x) \ \,\textrm{d} {x} \,\textrm{d} t , \nonumber \\&\qquad - \frac{1}{\varepsilon } \int _0^\tau \int _{{\overline{\Omega }}} \nabla _{x}\textbf{U}(t,x) : \textrm{d}{\mathfrak {R}}(t, \cdot ) \,\textrm{d} t - \frac{1}{\varepsilon ^2} \int _0^\tau \int _{\Omega } \textbf{U}(t,x) \cdot \left\langle {\mathcal {V}}_{t,x}; {\widetilde{\textbf{m}}}- {\widetilde{\varrho }}\textbf{U}(t,x) \right\rangle \ \,\textrm{d} {x} \,\textrm{d} t \nonumber \\&\qquad + \frac{1}{\varepsilon } \int _0^\tau \int _{\Omega } \left\langle {\mathcal {V}}_{t,x}; {\widetilde{\varrho }}\chi \left( \frac{{\widetilde{S}}}{{\widetilde{\varrho }}} \right) - {\widetilde{S}}\right\rangle \partial _S p(r, {\widehat{S}}) {\textrm{div}}_{x}\textbf{U}(t,x) \ \,\textrm{d} {x} \,\textrm{d} t \end{aligned}$$

(3.4)

for a.a. $\tau \in (0,T)$ and any trio of “test functions”

$$\begin{aligned}{} & {} r \in C^1([0,T] \times {\overline{\Omega }}),\ r > 0,\ {\widehat{S}} \in C^1([0,T] \times {\overline{\Omega }}),\nonumber \\{} & {} {\widehat{S}} = r s (r, \Theta ),\ \textbf{U}\in C^1([0,T] \times {\overline{\Omega }}; R^d), \nonumber \\{} & {} \textbf{U}\cdot \textbf{n}|_{\partial \Omega } = 0. \end{aligned}$$

(3.5)

4 Convergence, proof of Theorem 2.2

In view of hypothesis (2.12), we consider the following ansatz:

$$\begin{aligned} {\widehat{S}} = r s, \ \textbf{U}^\varepsilon = - \varepsilon \frac{1}{r} \nabla _{x}p(r, r s), \ \partial _t r + \frac{1}{\varepsilon } {\textrm{div}}_{x}(r \textbf{U}^\varepsilon ) = 0, \ \partial _t s - \frac{1}{\varepsilon } \textbf{U}^\varepsilon \cdot \nabla _{x}s = 0.\nonumber \\ \end{aligned}$$

(3.6)

As a consequence of (3.3), we get

$$\begin{aligned} \partial _t {\widehat{S}} + \frac{1}{\varepsilon } {\textrm{div}}_{x}( {\widehat{S}} \textbf{U}^\varepsilon )&=0, \nonumber \\ \partial _t \Theta + \frac{1}{\varepsilon } \textbf{U}^\varepsilon \cdot \nabla _{x}\Theta + \frac{1}{\varepsilon } \partial _S p(r, {\widehat{S}}) {\textrm{div}}_{x}\textbf{U}^\varepsilon&=0, \nonumber \end{aligned}$$

and

$$\begin{aligned} \left\langle {\mathcal {V}}^\varepsilon _{t,x}; {\widetilde{\varrho }}\textbf{U}^\varepsilon (t,x) - {\widetilde{\textbf{m}}}\right\rangle \frac{1}{\varepsilon r} \nabla _{x}p(r, {\widehat{S}}) - \frac{1}{\varepsilon ^2} \textbf{U}^\varepsilon (t,x) \cdot \left\langle {\mathcal {V}}^\varepsilon _{t,x}; {\widetilde{\textbf{m}}}- {\widetilde{\varrho }}\textbf{U}^\varepsilon (t,x) \right\rangle =0. \end{aligned}$$

Accordingly, the relative entropy inequality (3.5) simplifies considerably yielding

$$\begin{aligned}&\int _{\Omega } \left\langle {\mathcal {V}}_{\tau ,x}^\varepsilon ; E \left( {\widetilde{\varrho }}, {\widetilde{S}}, {\widetilde{\textbf{m}}}\ \Big |\ r, {\widehat{S}} , r \textbf{U}^\varepsilon \right) (\tau ,x) \right\rangle \ \,\textrm{d} {x} + \Lambda \int _{\Omega } \textrm{d}\ \textrm{trace} [ {\mathfrak {R}}^\varepsilon ](\tau , \cdot ) \nonumber \\&\qquad + \frac{1}{\varepsilon ^2} \int _0^\tau \int _{\Omega } \left\langle {\mathcal {V}}^\varepsilon _{t,x} ;{\widetilde{\varrho }}\left| \frac{{\widetilde{\textbf{m}}}}{{\widetilde{\varrho }}} - \textbf{U}^\varepsilon (t,x) \right| ^2 \right\rangle \ \,\textrm{d} {x} \,\textrm{d} t \nonumber \\&\quad \le \int _{\Omega } E \left( \varrho _{0,\varepsilon }, S_{0, \varepsilon } , \textbf{m}_{0,\varepsilon } \ \Big |\ r_0 , r_0 s_0 , r_0 \textbf{U}^\varepsilon (0,x) \right) \ \,\textrm{d} {x} \nonumber \\&\qquad - \frac{1}{\varepsilon } \int _0^\tau \int _{\Omega } \left\langle {\mathcal {V}}^\varepsilon _{t,x}; \mathbbm {1}_{{\widetilde{\varrho }}> 0} \frac{ ({\widetilde{\varrho }}\textbf{U}^\varepsilon (t,x) - {\widetilde{\textbf{m}}}) \otimes ({\widetilde{\varrho }}\textbf{U}^\varepsilon (t,x) - {\widetilde{\textbf{m}}}) }{{\widetilde{\varrho }}} : \nabla _{x}\textbf{U}^\varepsilon (t,x) \right\rangle \ \,\textrm{d} {x} \,\textrm{d} t \nonumber \\&\qquad - \frac{1}{\varepsilon } \int _0^\tau \int _{\Omega } \left\langle {\mathcal {V}}^\varepsilon _{t,x}; \left( p({\widetilde{\varrho }},{\widetilde{S}}) - ({\widetilde{\varrho }}- r) \partial _\varrho p(r, {\widehat{S}}) - ({\widetilde{S}}- {\widehat{S}}) \partial _S p(r,{\widehat{S}} ) - p(r, {\widehat{S}}) \right) (t,x) \right\rangle \nonumber \\&\qquad {\textrm{div}}_{x}\textbf{U}^\varepsilon (t,x) \ \,\textrm{d} {x} \,\textrm{d} t \nonumber \\&\qquad + \int _0^\tau \int _{\Omega } \left\langle {\mathcal {V}}^\varepsilon _{t,x}; {\widetilde{\varrho }}\textbf{U}^\varepsilon (t,x) - {\widetilde{\textbf{m}}}\right\rangle \Big ( \partial _t \textbf{U}^\varepsilon + \frac{1}{\varepsilon } \textbf{U}^\varepsilon \cdot \nabla _{x}\textbf{U}^\varepsilon \Big ) (t,x) \ \,\textrm{d} {x} \,\textrm{d} t \nonumber \\&\qquad + \frac{1}{\varepsilon } \int _0^\tau \int _{\Omega } \left\langle {\mathcal {V}}^\varepsilon _{t,x} ; \left( {\widetilde{\varrho }}\frac{{\widehat{S}}}{r}(t,x) - {\widetilde{\varrho }}\chi \left( \frac{{\widetilde{S}}}{{\widetilde{\varrho }}} \right) \right) \left( \frac{{\widetilde{\textbf{m}}}}{{\widetilde{\varrho }}} - \textbf{U}^\varepsilon (t,x) \right) \right\rangle \cdot \nabla _{x}\Theta (t,x) \ \,\textrm{d} {x} \,\textrm{d} t , \nonumber \\&\qquad - \frac{1}{\varepsilon } \int _0^\tau \int _{{\overline{\Omega }}} \nabla _{x}\textbf{U}^\varepsilon (t,x) : \textrm{d}{\mathfrak {R}}^\varepsilon (t, \cdot ) \,\textrm{d} t \nonumber \\&\qquad + \frac{1}{\varepsilon } \int _0^\tau \int _{\Omega } \left\langle {\mathcal {V}}^\varepsilon _{t,x}; {\widetilde{\varrho }}\chi \left( \frac{{\widetilde{S}}}{{\widetilde{\varrho }}} \right) - {\widetilde{S}}\right\rangle \partial _S p(r, {\widehat{S}}) {\textrm{div}}_{x}\textbf{U}^\varepsilon (t,x) \ \,\textrm{d} {x} \,\textrm{d} t . \end{aligned}$$

(4.1)

Next, using the ansatz for $\textbf{U}^\varepsilon$ from (4.1), we have

$$\begin{aligned} - \frac{1}{\varepsilon }&\int _0^\tau \int _{\Omega } \left\langle {\mathcal {V}}^\varepsilon _{t,x}; \mathbbm {1}_{{\widetilde{\varrho }}> 0} \frac{ ({\widetilde{\varrho }}\textbf{U}^\varepsilon (t,x) - {\widetilde{\textbf{m}}}) \otimes ({\widetilde{\varrho }}\textbf{U}^\varepsilon (t,x) - {\widetilde{\textbf{m}}}) }{{\widetilde{\varrho }}} : \nabla _{x}\textbf{U}^\varepsilon (t,x) \right\rangle \ \,\textrm{d} {x} \,\textrm{d} t \nonumber \\&\qquad - \frac{1}{\varepsilon } \int _{{\overline{\Omega }}} \int _0^\tau \nabla _{x}\textbf{U}^\varepsilon (t,x) : \textrm{d}{\mathfrak {R}}^\varepsilon (t, \cdot ) \,\textrm{d} t \nonumber \\&\quad {\mathop {\sim }\limits ^{<}}\int _0^\tau \left( \int _{\Omega } \left\langle {\mathcal {V}}_{t,x}^\varepsilon ; E \left( {\widetilde{\varrho }}, {\widetilde{S}}, {\widetilde{\textbf{m}}}\ \Big |\ r, {\widehat{S}} , r \textbf{U}^\varepsilon \right) (\tau ,x) \right\rangle \ \,\textrm{d} {x} + \Lambda \int _{\Omega } \textrm{d}\ \textrm{trace} [ {\mathfrak {R}}^\varepsilon ](t, \cdot ) \right) \,\textrm{d} t . \end{aligned}$$

(4.2)

Similarly, by virtue of hypothesis (2.11),

$$\begin{aligned}&- \frac{1}{\varepsilon } \int _0^\tau \int _{\Omega } \left\langle {\mathcal {V}}^\varepsilon _{t,x}; \left( p({\widetilde{\varrho }},{\widetilde{S}}) - ({\widetilde{\varrho }}- r) \partial _\varrho p(r, {\widehat{S}}) - ({\widetilde{S}}- {\widehat{S}}) \partial _S p(r,{\widehat{S}} ) - p(r, {\widehat{S}}) \right) (t,x) \right\rangle \nonumber \\&\qquad {\textrm{div}}_{x}\textbf{U}^\varepsilon (t,x) \ \,\textrm{d} {x} \,\textrm{d} t \nonumber \\&\quad {\mathop {\sim }\limits ^{<}}\int _0^\tau \int _{\Omega } \left\langle {\mathcal {V}}_{t,x}^\varepsilon ; E \left( {\widetilde{\varrho }}, {\widetilde{S}}, {\widetilde{\textbf{m}}}\ \Big |\ r, {\widehat{S}} , r \textbf{U}^\varepsilon \right) (\tau ,x) \right\rangle \ \,\textrm{d} {x} \,\textrm{d} t . \end{aligned}$$

(4.3)

The details of the above estimates follow the same lines as e.g. [9, Section 5.4]. Consequently, inequality (4.2) reduces to

$$\begin{aligned}&\int _{\Omega } \left\langle {\mathcal {V}}_{\tau ,x}^\varepsilon ; E \left( {\widetilde{\varrho }}, {\widetilde{S}}, {\widetilde{\textbf{m}}}\ \Big |\ r, {\widehat{S}} , r \textbf{U}^\varepsilon \right) (\tau ,x) \right\rangle \ \,\textrm{d} {x} + \Lambda \int _{\Omega } \textrm{d}\ \textrm{trace} [ {\mathfrak {R}}^\varepsilon ](\tau , \cdot ) \nonumber \\&\qquad + \frac{1}{\varepsilon ^2} \int _0^\tau \int _{\Omega } \left\langle {\mathcal {V}}^\varepsilon _{t,x} ;{\widetilde{\varrho }}\left| \frac{{\widetilde{\textbf{m}}}}{{\widetilde{\varrho }}} - \textbf{U}^\varepsilon (t,x) \right| ^2 \right\rangle \ \,\textrm{d} {x} \,\textrm{d} t \nonumber \\&\quad \le \int _{\Omega } E \left( \varrho _{0,\varepsilon }, S_{0, \varepsilon } , \textbf{m}_{0,\varepsilon } \ \Big |\ r_0 , r_0 s_0 , r_0 \textbf{U}^\varepsilon (0,x) \right) \ \,\textrm{d} {x} \nonumber \\&\qquad + \int _0^\tau \left( \int _{\Omega } \left\langle {\mathcal {V}}_{t,x}^\varepsilon ; E \left( {\widetilde{\varrho }}, {\widetilde{S}}, {\widetilde{\textbf{m}}}\ \Big |\ r, {\widehat{S}} , r \textbf{U}^\varepsilon \right) (\tau ,x) \right\rangle \ \,\textrm{d} {x} + \Lambda \int _{\Omega } \textrm{d}\ \textrm{trace} [ {\mathfrak {R}}^\varepsilon ](t, \cdot ) \right) \,\textrm{d} t \nonumber \\&\qquad + \int _0^\tau \int _{\Omega } \left\langle {\mathcal {V}}^\varepsilon _{t,x}; {\widetilde{\varrho }}\textbf{U}^\varepsilon (t,x) - {\widetilde{\textbf{m}}}\right\rangle \Big ( \partial _t \textbf{U}^\varepsilon + \frac{1}{\varepsilon } \textbf{U}^\varepsilon \cdot \nabla _{x}\textbf{U}^\varepsilon \Big ) (t,x) \ \,\textrm{d} {x} \,\textrm{d} t \nonumber \\&\qquad + \frac{1}{\varepsilon } \int _0^\tau \int _{\Omega } \left\langle {\mathcal {V}}^\varepsilon _{t,x} ; \left( {\widetilde{\varrho }}\frac{{\widehat{S}}}{r}(t,x) - {\widetilde{\varrho }}\chi \left( \frac{{\widetilde{S}}}{{\widetilde{\varrho }}} \right) \right) \left( \frac{{\widetilde{\textbf{m}}}}{{\widetilde{\varrho }}} - \textbf{U}^\varepsilon (t,x) \right) \right\rangle \cdot \nabla _{x}\Theta (t,x) \ \,\textrm{d} {x} \,\textrm{d} t , \nonumber \\&\qquad + \frac{1}{\varepsilon } \int _0^\tau \int _{\Omega } \left\langle {\mathcal {V}}^\varepsilon _{t,x}; {\widetilde{\varrho }}\chi \left( \frac{{\widetilde{S}}}{{\widetilde{\varrho }}} \right) - {\widetilde{S}}\right\rangle \partial _S p(r, {\widehat{S}}) {\textrm{div}}_{x}\textbf{U}^\varepsilon (t,x). \ \,\textrm{d} {x} \,\textrm{d} t \end{aligned}$$

(4.4)

By virtue of our choice of the initial data (2.12),

$$\begin{aligned} \int _{\Omega } E \left( \varrho _{0,\varepsilon }, S_{0, \varepsilon }, \textbf{m}_{0,\varepsilon } \ \Big |\ r_0, r_0 s_0, r_0 \textbf{U}^\varepsilon (0,x) \right) \ \,\textrm{d} {x} \rightarrow 0 \ \text{ as }\ \varepsilon \rightarrow 0. \end{aligned}$$

Moreover, as E is strictly convex, the energy inequality (2.6) implies boundedness of the first moments

$$\begin{aligned} \left\langle {\mathcal {V}}^\varepsilon ; {\widetilde{\varrho }}\right\rangle ,\ \left\langle {\mathcal {V}}^\varepsilon ; |{\widetilde{\textbf{m}}}| \right\rangle \ \text{ in }\ L^\infty (0,T; L^1(\Omega )). \end{aligned}$$

(4.5)

Consequently, we deduce from (4.5)

$$\begin{aligned}&\int _{\Omega } \left\langle {\mathcal {V}}_{\tau ,x}^\varepsilon ; E \left( {\widetilde{\varrho }}, {\widetilde{S}}, {\widetilde{\textbf{m}}}\ \Big |\ r, {\widehat{S}} , r \textbf{U}^\varepsilon \right) (\tau ,x) \right\rangle \ \,\textrm{d} {x} + \Lambda \int _{\Omega } \textrm{d}\ \textrm{trace} [ {\mathfrak {R}}^\varepsilon ](\tau , \cdot ) \nonumber \\&\qquad + \frac{1}{\varepsilon ^2} \int _0^\tau \int _{\Omega } \left\langle {\mathcal {V}}^\varepsilon _{t,x} ;{\widetilde{\varrho }}\left| \frac{{\widetilde{\textbf{m}}}}{{\widetilde{\varrho }}} - \textbf{U}^\varepsilon (t,x) \right| ^2 \right\rangle \ \,\textrm{d} {x} \,\textrm{d} t \nonumber \\&\quad \le \int _0^\tau \left( \int _{\Omega } \left\langle {\mathcal {V}}_{t,x}^\varepsilon ; E \left( {\widetilde{\varrho }}, {\widetilde{S}}, {\widetilde{\textbf{m}}}\ \Big |\ r, {\widehat{S}} , r \textbf{U}^\varepsilon \right) \right\rangle \ \,\textrm{d} {x} + \Lambda \int _{\Omega } \textrm{d}\ \textrm{trace} [ {\mathfrak {R}}^\varepsilon ](t, \cdot ) \right) \,\textrm{d} t \nonumber \\&\qquad + \frac{1}{\varepsilon } \int _0^\tau \int _{\Omega } \left\langle {\mathcal {V}}^\varepsilon _{t,x} ; \left( {\widetilde{\varrho }}\frac{{\widehat{S}}}{r}(t,x) - {\widetilde{\varrho }}\chi \left( \frac{{\widetilde{S}}}{{\widetilde{\varrho }}} \right) \right) \left( \frac{{\widetilde{\textbf{m}}}}{{\widetilde{\varrho }}} - \textbf{U}^\varepsilon (t,x) \right) \right\rangle \cdot \nabla _{x}\Theta (t,x) \ \,\textrm{d} {x} \,\textrm{d} t , \nonumber \\&\qquad + \frac{1}{\varepsilon } \int _0^\tau \int _{\Omega } \left\langle {\mathcal {V}}^\varepsilon _{t,x}; {\widetilde{\varrho }}\chi \left( \frac{{\widetilde{S}}}{{\widetilde{\varrho }}} \right) - {\widetilde{S}}\right\rangle \partial _S p(r, {\widehat{S}}) {\textrm{div}}_{x}\textbf{U}^\varepsilon (t,x) \ \,\textrm{d} {x} \,\textrm{d} t + h(\varepsilon ), \end{aligned}$$

(4.6)

where $h(\varepsilon ) \rightarrow 0$ as $\varepsilon \rightarrow 0$. Choosing

$$\begin{aligned} \chi (Z) = Z \ \text{ for }\ Z \le M, \ \text{ where }\ M \ \text{ is } \text{ large } \text{ enough }, \end{aligned}$$

we get

$$\begin{aligned} \frac{1}{\varepsilon }&\int _0^\tau \int _{\Omega } \left\langle {\mathcal {V}}^\varepsilon _{t,x} ; \left( {\widetilde{\varrho }}\frac{{\widehat{S}}}{r}(t,x) - {\widetilde{\varrho }}\chi \left( \frac{{\widetilde{S}}}{{\widetilde{\varrho }}} \right) \right) \left( \frac{{\widetilde{\textbf{m}}}}{{\widetilde{\varrho }}} - \textbf{U}^\varepsilon (t,x) \right) \right\rangle \cdot \nabla _{x}\Theta (t,x) \ \,\textrm{d} {x} \,\textrm{d} t \nonumber \\&\le \frac{1}{2\varepsilon ^2} \int _0^\tau \int _{\Omega } \left\langle {\mathcal {V}}^\varepsilon _{t,x} ;{\widetilde{\varrho }}\left| \frac{{\widetilde{\textbf{m}}}}{{\widetilde{\varrho }}} - \textbf{U}^\varepsilon (t,x) \right| ^2 \right\rangle \ \,\textrm{d} {x} \,\textrm{d} t \nonumber \\&\quad + c \int _0^\tau \int _{\Omega } \left\langle {\mathcal {V}}^\varepsilon _{t,x} ; {\widetilde{\varrho }}\left| \chi \left( \frac{{\widehat{S}}}{r} \right) (t,x)- \chi \left( \frac{{\widetilde{S}}}{{\widetilde{\varrho }}} \right) \right| ^2 \right\rangle \ \,\textrm{d} {x} \,\textrm{d} t , \nonumber \end{aligned}$$

where

$$\begin{aligned}{} & {} \int _0^\tau \int _{\Omega } \left\langle {\mathcal {V}}^\varepsilon _{t,x}; {\widetilde{\varrho }}\left| \chi \left( \frac{{\widehat{S}}}{r} \right) (t,x)- \chi \left( \frac{{\widetilde{S}}}{{\widetilde{\varrho }}} \right) \right| ^2 \right\rangle \ \,\textrm{d} {x} \,\textrm{d} t \nonumber \\{} & {} \quad {\mathop {\sim }\limits ^{<}}\int _0^\tau \int _{\Omega } \left\langle {\mathcal {V}}_{t,x}^\varepsilon ; E \left( {\widetilde{\varrho }}, {\widetilde{S}}, {\widetilde{\textbf{m}}}\ \Big |\ r, {\widehat{S}}, r \textbf{U}^\varepsilon \right) \right\rangle \ \,\textrm{d} {x}\,\textrm{d} t . \end{aligned}$$

(4.7)

Similarly

$$\begin{aligned} \frac{1}{\varepsilon } \int _0^\tau \int _{\Omega } \left\langle {\mathcal {V}}^\varepsilon _{t,x}; {\widetilde{\varrho }}\chi \left( \frac{{\widetilde{S}}}{{\widetilde{\varrho }}} \right) - {\widetilde{S}}\right\rangle \partial _S p(r, {\widehat{S}}) {\textrm{div}}_{x}\textbf{U}^\varepsilon (t,x) \ \,\textrm{d} {x} \,\textrm{d} t \\ {\mathop {\sim }\limits ^{<}}\int _0^\tau \int _{\Omega } \left\langle {\mathcal {V}}_{t,x}^\varepsilon ; E \left( {\widetilde{\varrho }}, {\widetilde{S}}, {\widetilde{\textbf{m}}}\ \Big |\ r, {\widehat{S}} , r \textbf{U}^\varepsilon \right) \right\rangle \ \,\textrm{d} {x} \,\textrm{d} t . \end{aligned}$$

Thus (4.7) reduces to

$$\begin{aligned}&\int _{\Omega } \left\langle {\mathcal {V}}_{\tau ,x}^\varepsilon ; E \left( {\widetilde{\varrho }}, {\widetilde{S}}, {\widetilde{\textbf{m}}}\ \Big |\ r, {\widehat{S}} , r \textbf{U}^\varepsilon \right) (\tau ,x) \right\rangle \ \,\textrm{d} {x} + \Lambda \int _{\Omega } \textrm{d}\ \textrm{trace} [ {\mathfrak {R}}^\varepsilon ](\tau , \cdot ) \nonumber \\&\qquad + \frac{1}{2\varepsilon ^2} \int _0^\tau \int _{\Omega } \left\langle {\mathcal {V}}^\varepsilon _{t,x} ;{\widetilde{\varrho }}\left| \frac{{\widetilde{\textbf{m}}}}{{\widetilde{\varrho }}} - \textbf{U}^\varepsilon (t,x) \right| ^2 \right\rangle \ \,\textrm{d} {x} \,\textrm{d} t \nonumber \\&\quad \le \int _0^\tau \left( \int _{\Omega } \left\langle {\mathcal {V}}_{t,x}^\varepsilon ; E \left( {\widetilde{\varrho }}, {\widetilde{S}}, {\widetilde{\textbf{m}}}\ \Big |\ r, {\widehat{S}} , r \textbf{U}^\varepsilon \right) \right\rangle \ \,\textrm{d} {x} + \Lambda \int _{\Omega } \textrm{d}\ \textrm{trace} [ {\mathfrak {R}}^\varepsilon ](t, \cdot ) \right) \,\textrm{d} t + h(\varepsilon ). \nonumber \end{aligned}$$

(4.8)

Consequently, the conclusion of Theorem 2.2 follows by a direct application of Gronwall’s lemma and the bounds (4.6).