Dimension reduction for compressible Navier–Stokes equations with density-dependent viscosity

Abstract

In this paper, we investigate the Navier–Stokes equations describing the motion of a compressible viscous fluid confined to a thin domain $\varOmega _{\varepsilon }=I_{\varepsilon }\times (0, 1)$, $I_{ \varepsilon }=(0, \varepsilon )\subset \mathbb{R}$. We show that the strong solutions in the 2D domain converge to the classical solutions of the limit 1D Navier–Stokes system as $\varepsilon \to 0$.

1 Introduction

We consider the Navier–Stokes system for a barotropic compressible viscous fluid which in the case of two-dimensional motion has the form [10, 15, 20]

$$ \textstyle\begin{cases} \partial _{t}\rho _{\varepsilon }+\operatorname{div}_{x}(\rho _{\varepsilon } u_{ \varepsilon })=0, \\ \partial _{t}(\rho _{\varepsilon } u_{\varepsilon })+\operatorname{div}_{x}( \rho _{\varepsilon } u_{\varepsilon }\otimes u_{\varepsilon })+ \nabla _{x}P(\rho _{\varepsilon })=\operatorname{div}_{x}\mathbb{S}(\nabla _{x}u_{ \varepsilon }), \end{cases} $$

(1.1)

where $\rho _{\varepsilon }=\rho _{\varepsilon }(x, t)$ and $u_{\varepsilon }=(u_{\varepsilon }^{1}(x, t), u_{\varepsilon }^{2}(x, t))$ stand for the unknown fluid mass density and the velocity field, respectively, and the viscous stress tensor $\mathbb{S}$ is a linear function of the velocity gradient and therefore described by the Newton law:

$$ \mathbb{S}(\nabla _{x}u_{\varepsilon })=\mu \bigl(\nabla _{x}u_{ \varepsilon }+\nabla _{x}^{t} u_{\varepsilon }\bigr)+\lambda (\rho _{ \varepsilon })\operatorname{div}_{x}u_{\varepsilon } \mathbb{I}, $$

the shear viscosity coefficient μ, the bulk one λ and the pressure P are defined on $(0, +\infty )$ and satisfy the conditions

$$ 0< \mu =\mathrm{const}, \qquad \lambda (\rho _{\varepsilon })=b \rho _{ \varepsilon }^{\beta }, \qquad P(\rho _{\varepsilon })=a \rho _{ \varepsilon }^{\gamma } $$

for positive constants $a>0$, $b>0$. In the sequel, we set $a=b=1$ without loss of generality, and we also have the following hypotheses on γ and β:

$$ \beta >3, \qquad \gamma > 1. $$

The fluid is confined to a bounded physical domain $\varOmega _{\varepsilon }\subset \mathbb{R}^{2}$, on the boundary of which we impose the complete slip boundary conditions

$$ u_{\varepsilon }\cdot n|_{\partial \varOmega _{\varepsilon }}=0, \qquad \bigl[\mathbb{S}( \nabla _{x}u_{\varepsilon })\cdot n \bigr]\times n |_{\partial \varOmega _{\varepsilon }}=0, $$

(1.2)

where the symbol n denotes the outer normal vector. The motion originates from the initial state

$$ \rho _{\varepsilon }(\cdot , 0)=\rho _{0, \varepsilon },\qquad u_{ \varepsilon }(\cdot , 0)=u_{0, \varepsilon }, \quad x\in \varOmega _{ \varepsilon }. $$

(1.3)

We remark that the use of the slip instead of the more conventional no-slip condition $u_{\varepsilon }|_{\partial \varOmega _{\varepsilon }}=0$ is quite natural in the present context as the latter would completely stop the fluid motion in the asymptotic limit $\varepsilon \to 0$.

Although all fluid flows are in general two-dimensional, in many cases the specific shape of the physical domain enforce major changes in the density and velocity only in one direction. A typical example is the fluid flow confined to a thin domain that can be effectively described by using only spatial variable. We consider a family of shrinking domains:

$$ \varOmega _{\varepsilon }=I_{\varepsilon }\times (0, 1),\qquad I_{ \varepsilon } \subset \mathbb{R},\qquad I_{\varepsilon }=\varepsilon I \subset \mathbb{R},\quad \varepsilon \to 0, $$

where $I \subset \mathbb{R}$ is a unit interval. Under suitable conditions on the initial data it is natural to expect that the strong solution $(\rho _{\varepsilon }, u_{\varepsilon })$ of (1.1)–(1.2) on $\varOmega _{\varepsilon }$ tends, as $\varepsilon \to 0$, to a classical solution $(\rho , u )$ of the 1D system on $(0, 1)$:

$$ \textstyle\begin{cases} \partial _{t}\rho +\partial _{y}(\rho u)=0, \\ \partial _{t}(\rho u)+\partial _{y} (\rho u^{2}+P(\rho ) )= \partial _{y} (\nu (\rho )\partial _{y}u ), \qquad \nu (\rho )=2 \mu +\lambda (\rho ). \end{cases} $$

(1.4)

The boundary conditions (1.2) naturally lead to the no-slip boundary conditions for the velocity, i.e.,

$$ u(0, \cdot )=u(1, \cdot )=0. $$

(1.5)

Since we are interested in smooth solutions of the 1D equations, we complement the system of equations (1.4) and (1.5) with the initial conditions

$$ \rho (\cdot , 0)=\rho _{0},\qquad u(\cdot , 0)=u_{0}, \quad x\in (0, 1). $$

(1.6)

Hereinafter we use the notation $x=(x_{1}, y)\in \mathbb{R}^{2}$, $y\in \mathbb{R}$ and denote the derivative in $x_{2}$ by $\partial _{y}$. In this paper we give a rigorous justification of the convergence $(\rho _{\varepsilon }, u_{\varepsilon })\to (\rho , u)$ as $\varepsilon \to 0$.

As far as we know, the limit passage for fluid flows has not yet been rigorously investigated and there is only a handful of results on related problems. Since incompressibility in one dimension does not allow for any movement, such a limit makes little sense for 1D incompressibility flows. However, dimension reduction to 2D-planar flows was examined in [11, 17–19]; see also the references given therein.

The case of a compressible barotropic fluid was studied by many authors. Vodák in [22] studied the steady and nonsteady Navier–Stokes system for barotropic compressible flow. For three-dimensional system, Bella, Feireisl and Novotný in [1] considered the motion of a compressible viscous fluid confined to a cavity shaped as a thin rod $\varOmega _{\varepsilon }=\varepsilon Q\times (0, 1)$, $Q\subset \mathbb{R}^{2}$, they showed that the weak solutions in the 3D domain converge to (strong) solutions of the limit 1D Navier–Stokes system as $\varepsilon \to 0$. Březina–Kreml–Mácha in [2] studied the dimension reduction for the full Navier–Stokes–Fourier system in a thin long pipe $\varOmega _{\varepsilon }=\varepsilon Q\times (0, 1)\in \mathbb{R}^{3}$, where Q is an open rectangular domain in $\mathbb{R}^{2}$, they showed that the weak solutions of 3D system on $\varOmega _{\varepsilon }$ tend, as $\varepsilon \to 0$, to a classical solution of 1D system on $(0, 1)$. Ducomet–Caggio–Nečasová–Pokorný in [5] investigated the rotating Navier–Stokes–Fourier–Poisson system confined to a straight layer $\varOmega _{\varepsilon }=\omega \times (0, \varepsilon )$, where ω is a 2-D domain, they showed that the weak solutions in the 3d domain converge to the strong solution of the 2-D system on $\omega \to 0$ as $\varepsilon \to 0$ on the time interval, where the strong solution exists.

Motivated by [1, 5] and [2], our main purpose in this paper is to show that the strong solution of 2D compressible Navier–Stokes system confined to a thin domain $\varOmega _{\varepsilon }=(0, \epsilon )\times (0, 1)$ converge to the classical solution of the 1D Navier–Stokes system on $(0, 1)$ as $\varepsilon \to 0$.

In elasticity theory, the analysis of similar dimension reduction problems leans on variants of the Korn inequality which controls the gradient of velocity v by its symmetric part, specifically,

$$ \Vert \nabla _{x}v \Vert _{L^{2}(\varOmega _{\varepsilon })} \leqslant c( \varepsilon ) \bigl\Vert \nabla _{x}v+\nabla _{x}^{t}v \bigr\Vert _{L^{2}(\varOmega _{ \varepsilon })}, \qquad v\cdot n|_{\partial \varOmega _{\varepsilon }}=0. $$

(1.7)

Two problems have arisen in (1.7). Firstly, the kernel of the linear operator $v\mapsto \nabla _{x}v+\nabla _{x}^{t}v$, $v\cdot n|_{\partial \varOmega _{\varepsilon }}$ has to be empty, in particular, the “bottom” set I must not be rotationally symmetric. Secondly, for any fixed $\varepsilon >0$, even if (1.7) holds, the constant $c(\varepsilon )$ blows up for $\varepsilon \to 0$ unless some necessary restrictions are imposed on the field v, and this is true even if the set I is not rotationally symmetric, cf. the interesting paper by Lewicka and Müller in [14].

Bella, Feireisl and Novotný in [1] obtained their result for a regular planar domain since they avoid the use of Kron’s inequality by exploring the structural stability of the family of solutions of the barotropic Navier–Stokes system. It is not difficult to see that the problems arising in the context of compressible fluids would need a stronger analogue of (1.7), namely

$$ \Vert \nabla _{x}v \Vert _{L^{2}(\varOmega _{\varepsilon })} \leqslant c( \varepsilon ) \biggl\Vert \nabla _{x}v+\nabla _{x}^{t}v-\frac {2}{3}\operatorname{div}_{x}v \mathbb{I} \biggr\Vert _{L^{2}(\varOmega _{\varepsilon })},\qquad v\cdot n|_{ \partial \varOmega _{\varepsilon }}=0. $$

(1.8)

In view of the above-mentioned difficulties related to the validity of (1.7) or (1.8), our approach relies on the structural stability of the family of solutions of the barotropic Navier–Stokes system encoded in the relative entropy inequality introduced in [6, 8]. This method is basically independent of the specific form of the viscous stress and of possible “dissipative” bounds for the Navier–Stokes system.

The paper is organized as follows. In Sect. 2 we introduce the relative entropy inequality and formulate our main result. In Sect. 3, we establish convergence towards the target system (1.4).

2 Preliminaries, main result

In this section, we first introduce the relative entropy inequality, and then give the solutions of the target systems (1.1) and (1.4), finally, we state our main result.

2.1 Relative entropy inequality

The proof of our main theorem is based on the method of the relative entropy (see [8], Dafermos in [3], Germain in [9] and Mellet, Vasseur in [16]). The relative entropy$\mathcal{E}_{\varepsilon } ([\rho , u]| [r, U] )$ with respect to $[r, U]$ is defined as

$$ \mathcal{E}_{\varepsilon } \bigl([\rho , u]\big| [r, U] \bigr) = \frac {1}{ \vert I_{ \varepsilon } \vert } \int _{0}^{1} \int _{I_{\varepsilon }} \biggl(\frac {1}{2} \rho \vert u-U \vert ^{2}+H(\rho )-H'(r) (\rho -r)-H(r) \biggr) \,dx_{1}\,dx_{2}, $$

(2.1)

where the potential $H(\rho )$ is defined (modulo a linear function) through

$$ H(\rho )=\rho \int _{\tilde{\rho }}^{\rho }\frac {P(s)}{s^{2}} \,ds, $$

then

$$ H'(\rho )\rho -H(\rho )=P(\rho ), $$

(2.2)

along with the relative entropy inequality

$$ \begin{aligned}[b] &\mathcal{E}_{\varepsilon } \bigl([ \rho , u]\big| [r, U] \bigr) (t) +\frac {1}{ \vert I_{\varepsilon } \vert } \int _{0}^{\tau } \int _{\varOmega _{ \varepsilon }}\mathbb{S} \bigl(\nabla _{x}(u-U) \bigr): \nabla _{x}(u-U)\,dx \,dt \\ &\quad \leqslant \mathcal{E}_{\varepsilon } \bigl([\rho , u]\big| [r, U] \bigr) (0)+ \int _{0}^{\tau }\mathcal{F}_{\varepsilon }(\rho , u, r, U)\,dt, \end{aligned} $$

(2.3)

and the remainder $\mathcal{F}_{\varepsilon }$ reads

$$ \begin{aligned}[b] \mathcal{F}_{\varepsilon }(\rho , u, r, U) &=\frac {1}{ \vert I_{ \varepsilon } \vert } \int _{\varOmega _{\varepsilon }}\rho (U_{t}+u\cdot \nabla _{x} U)\cdot (U-u)\,dx \\ &\quad {}+\frac {1}{ \vert I_{\varepsilon } \vert } \int _{\varOmega _{\varepsilon }}r(U_{t}+U \cdot \nabla _{x} U)\cdot (u-U)\,dx \\ &\quad {}+\frac {1}{ \vert I_{\varepsilon } \vert } \int _{\varOmega _{\varepsilon }} \bigl[P(r)-P( \rho )+(\rho -r)P'(r) \bigr] \operatorname{div}_{x}U \,dx \\ &\quad {}+\frac {1}{ \vert I_{\varepsilon } \vert } \int _{\varOmega _{\varepsilon }} (\rho -r) (u-U) \cdot \nabla _{x}H'(r) \,dx. \end{aligned} $$

(2.4)

Here, the functions r, U are arbitrary smooth, r strictly positive, and U satisfying the no-slip boundary conditions (1.5). It is easy to check that (2.3) is satisfied as an equality as soon as solution ρ, u is smooth enough.

2.2 Solutions of target systems (1.1) and (1.4)

The existence of global-in-time strong solutions to the two-dimensional Navier–Stokes system (1.1) with complete slip boundary conditions was established by Vaigant and Kazhikhov in [21]. It reads as follows.

Proposition 2.1

Let$\varOmega \subset \mathbb{R}^{2}$be a rectangular domain. Assume that

$$ 0< \mu =\mathrm{const}, \qquad \lambda (\rho )= \rho ^{\beta },\qquad P(\rho )= \rho ^{\gamma } \quad \textit{for } \beta >3, \gamma >1. $$

(2.5)

If the initial data$(\rho _{0}, u_{0})$are such that

$$ \textstyle\begin{cases} \rho _{0}(x)\in W^{1, q}(\varOmega ), \qquad u_{0}(x)\in H^{2}(\varOmega ),\quad q>2, \\ 0< m_{0}\leqslant \rho _{0}\leqslant M_{0}< +\infty , \end{cases} $$

where$m_{0}$and$M_{0}$are some positive constants, and the agreement conditions

$$ \textstyle\begin{cases} u_{0}^{1}=\partial _{1}u_{0}^{2}=0 \quad \textit{on } \{x\in \varOmega | x_{1}=0, \varepsilon \}, \\ u_{0}^{2}=\partial _{2}u_{0}^{1}=0 \quad \textit{on } \{x\in \varOmega | x_{2}=0, 1\}, \\ [\partial _{1}(2\mu \partial _{1}u_{0}^{1})+\partial _{1}(\lambda ( \rho _{0})(\partial _{1}u_{0}^{1}+\partial _{2}u_{0}^{2}))+\partial _{2}( \mu (\partial _{2}u_{0}^{2}+\partial _{1}u_{0}^{2}))-\partial _{1}(P( \rho _{0}))]|_{x_{1}=0, \varepsilon } =0, \\ [\partial _{2}(2\mu \partial _{2}u_{0}^{2})+\partial _{2}(\lambda ( \rho _{0})(\partial _{1}u_{0}^{1}+\partial _{2}u_{0}^{2}))+\partial _{1}( \mu (\partial _{2}u_{0}^{2}+\partial _{1}u_{0}^{2}))-\partial _{2}(P( \rho _{0}))]|_{x_{2}=0, 1} =0, \end{cases} $$

are satisfied for all$x\in \bar{\varOmega }$, where$\partial _{i}$denotes$\partial _{x_{i}}$, $i=1, 2$, then there exists a unique global strong solution$(\rho , u)$to problem (1.1)–(1.2) in$\varOmega \times (0, \infty )$satisfying

$$ \textstyle\begin{cases} \rho \in L^{q}(0, T; W^{1, q}(\varOmega )), \quad \partial _{t}\rho \in L^{ \infty }(0, T; L^{\infty }(\varOmega )), \\ u\in L^{2}(0, T; H^{2}(\varOmega )), \quad \partial _{t}u\in L^{2}(0, T; L^{2}( \varOmega )), \end{cases} $$

for any$0< T<\infty$.

Remark 2.1

From [1, 21] and [6], we know that, for two-dimensional compressible barotropic Navier–Stokes system, the global-in-time solution $(\rho _{\varepsilon }, u_{\varepsilon })$ of Eqs. (1.1) enjoys the regularity specified in Proposition 2.1, the relative entropy inequality in (2.3) is satisfied for $\rho =\rho _{\varepsilon }$, $u=u_{\varepsilon }$ and any pair of the test functions

$$ r\in C^{\infty }\bigl([0, T]\times \bar{\varOmega }_{\varepsilon } \bigr),\qquad r>0, \qquad U\in C^{\infty }\bigl([0, T]\times \bar{\varOmega }_{\varepsilon } \bigr),\qquad U\cdot n|_{\partial \varOmega _{\varepsilon }}=0, $$

and by means of density arguments, the class of test functions can be extended to less regular $(r, U)$.

For one-dimensional compressible Navier–Stokes system (1.4)–(1.6) with no-slip boundary conditions, it has been discussed by many mathematicians. Kazhikhov and Shelukhi in [13] (for polytropic perfect gas with constant viscosity) and Kawohl in [12] (for real gas with $\mu =\mu (\rho )$) got global classical solutions for large initial data with $\inf \rho _{0}>0$, respectively. Ding, Wen and Zhu in [4] obtained the global existence of classical solutions to the compressible Navier–Stokes equations in 1D when the initial data are large and the density dependent viscosity. For completeness, we state the proposition as follows.

Proposition 2.2

Let$\varOmega \in \mathbb{R}$be an unit interval. Assume that (2.5) holds, and the initial data$(\rho _{0}, u_{0})$satisfying

$$ \inf \rho _{0}>0, \qquad \rho _{0}\in H^{2}( \varOmega ), \qquad u_{0}\in H^{3}( \varOmega )\cap H_{0}^{1}(\varOmega ) $$

and the agreement condition

$$ \partial _{y} \bigl[\nu (\rho _{0})\partial _{y} u_{0} \bigr](y)- \partial _{y} \bigl[P(\rho _{0})\bigr](y)=\rho _{0}(y)g(y),\quad y\in [0, 1], $$

for a given function$g\in L^{2}(\varOmega )$. Then, for any$T>0$, there exists a unique global classical solution$(\rho , u)$to (1.4)–(1.6) satisfying

$$\begin{aligned}& \bigl(\rho , \rho ^{\gamma }\bigr)\in C\bigl([0, T]; H^{2}( \varOmega )\bigr), \qquad \bigl(\rho _{t}, \bigl(\rho ^{\gamma } \bigr)_{t}\bigr)\in C\bigl([0, T]; H^{1}(\varOmega )\bigr), \\& \rho _{tt}\in C\bigl([0, T]; L^{2}(\varOmega )\bigr),\qquad \bigl(\rho ^{\gamma }\bigr)_{tt} \in L^{\infty }\bigl([0, T]; L^{2}(\varOmega )\bigr), \qquad (\rho u)_{t}\in C\bigl([0, T]; H^{1}(\varOmega )\bigr), \\& u \in C\bigl([0, T]; H^{3}(\varOmega )\cap H_{0}^{1}( \varOmega )\bigr), \qquad u_{t} \in L^{\infty }\bigl([0, T]; H_{0}^{1}(\varOmega )\bigr)\cap L^{2}\bigl([0, T]; H^{2}( \varOmega )\bigr). \end{aligned}$$

2.3 Main result

We are ready to state our main result.

Theorem 2.1

Let$I_{\varepsilon }=(0, \varepsilon )\subset \mathbb{R}$and$\varOmega _{\varepsilon }=I_{\varepsilon }\times (0, 1)$for$\varepsilon >0$. Suppose that the system (1.1)–(1.2) admits a strong solution$(\rho _{\varepsilon }, u_{\varepsilon })$in$\varOmega _{\varepsilon } \times (0, T)$which emanating from the initial data$(\rho _{0, \varepsilon }, u_{0, \varepsilon })$, and the system (1.4)–(1.6) possesses a classical solution$(\rho , u)$in$(0, 1)\times (0, T)$emanating from$(\rho _{0}, u_{0})$.

Let

$$ \frac {1}{ \vert I_{\varepsilon } \vert } \int _{I_{\varepsilon }}\rho _{0, \varepsilon }(x_{1}, \cdot ) \,dx_{1}\to \rho _{0}, \qquad \frac {1}{ \vert I_{ \varepsilon } \vert } \int _{I_{\varepsilon }}\rho _{0, \varepsilon }u_{0, \varepsilon }(x_{1}, \cdot ) \,dx_{1}\to \rho _{0}v_{0}, $$

(2.6)

weakly in$L^{1}(0, 1)$, where$\inf \rho _{0}>0$, $v_{0}=(0, u_{0})$, and let

$$ \frac {1}{ \vert I_{\varepsilon } \vert } \int _{\varOmega _{\varepsilon }} \biggl[ \frac {1}{2}\rho _{0, \varepsilon } \vert u_{0, \varepsilon } \vert ^{2}+H(\rho _{0, \varepsilon }) \biggr]\,dx\to \int _{0}^{1} \biggl[\frac {1}{2}\rho _{0} \vert v_{0} \vert ^{2}+H( \rho _{0}) \biggr]\,dy, $$

(2.7)

where$H(\cdot )$defined as in (2.2). Then

$$\begin{aligned}& \operatorname{ess}\sup_{t\in (0, T)} \frac {1}{ \vert I_{\varepsilon } \vert } \Vert \rho _{ \varepsilon }-\rho \Vert ^{\gamma }_{L^{\gamma }(\varOmega _{\varepsilon })} \to 0, \end{aligned}$$

(2.8)

$$\begin{aligned}& \frac {1}{ \vert I_{\varepsilon } \vert } \bigl\Vert u_{\varepsilon }^{2}-u \bigr\Vert _{L^{2}( \varOmega _{\varepsilon }\times (0, T))}^{2} \to 0, \end{aligned}$$

(2.9)

as$\varepsilon \to 0$,where$u_{\varepsilon }=(u^{1}_{\varepsilon }, u^{2}_{\varepsilon })$.

3 Proof of Theorem 2.1

Let $\underline{\rho }$, $\overline{\rho }>0$ be two positive constants such that

$$ 0< \underline{\rho }\leqslant \frac {1}{2}\inf_{(y, t)\in (0, 1) \times (0, T)} \rho (y, t)\leqslant 2\sup_{(y, t)\in (0, 1)\times (0, T)} \rho (y, t)\leqslant \overline{\rho }. $$

Now, we introduce the set of essential values $\mathcal{O}_{\mathrm{ess}}\subset (0, \infty )$

$$ \mathcal{O}_{\mathrm{ess}}\triangleq \{ \rho _{\epsilon }\in \mathbb{R} | \underline{\rho }\leqslant \rho _{\varepsilon }\leqslant \overline{\rho } \}, $$

together with its residual counterpart

$$ \mathcal{O}_{\mathrm{res}}\triangleq (0, \infty ) \setminus \mathcal{O}_{\mathrm{ess}}. $$

Then each measurable function h can be written as

$$ h=h_{\mathrm{ess}}+h_{\mathrm{res}}, $$

where

$$ h_{\mathrm{ess}}= \textstyle\begin{cases} h(x, t) &\mbox{if } \rho _{\varepsilon }\in \mathcal{O}_{\mathrm{ess}}, \\ 0 &\mbox{otherwise}. \end{cases} $$

We have the following lemmas.

Lemma 3.1

Let$\underline{\rho }$, $\overline{\rho }>0$be two positive constants and let

$$ \mathcal{H}(\rho _{\varepsilon })\triangleq H(\rho _{\varepsilon })-H'( \rho ) (\rho _{\varepsilon }-\rho )-H(\rho ). $$

Then there exists some constant$c>0$independent of$\rho _{\varepsilon }$such that

$$\begin{aligned}& \mathcal{H}(\rho _{\varepsilon })\geqslant c \vert \rho _{ \varepsilon }- \rho \vert ^{2} \rho _{\varepsilon }\in [ \underline{\rho }, \overline{\rho }], \\& \mathcal{H}(\rho _{\varepsilon })\geqslant c \vert \rho _{\varepsilon }- \rho \vert \rho _{\varepsilon }< \underline{\rho }, \\& \mathcal{H}(\rho _{\varepsilon })\geqslant c\rho _{\varepsilon } \rho _{\varepsilon }>\overline{\rho }. \end{aligned}$$

The proof of the lemma can be found in [2, 7] and is therefore omitted here for simplicity.

Lemma 3.2

Under the conditions of Lemma 3.1, there exists some positive constant$c>0$independent of$\rho _{\varepsilon }$such that

$$ \mathcal{H}(\rho _{\varepsilon })\geqslant c\bigl(1+\rho _{\varepsilon }^{ \gamma } \bigr) \quad \textit{for } \rho _{\varepsilon }\in \mathcal{O}_{\mathrm{res}}. $$

Proof

By the definition of $H(\rho _{\varepsilon })$, we have

$$ H(\rho _{\varepsilon })=\rho _{\varepsilon } \int _{\underline{\rho }}^{ \rho _{\varepsilon }}\frac {P(s)}{s^{2}}\,ds = \frac {1}{\gamma -1}\bigl(\rho _{ \varepsilon }^{\gamma }-\rho _{\varepsilon }\underline{\rho }^{ \gamma -1}\bigr). $$

Then

$$ \begin{aligned} \mathcal{H}(\rho _{\varepsilon }) &=H(\rho _{ \varepsilon })-H'(\rho ) (\rho _{\varepsilon }-\rho )-H(\rho ) \\ & \geqslant \frac {1}{\gamma -1}\bigl(\rho _{\varepsilon }^{\gamma }- \rho _{ \varepsilon }\underline{\rho }^{\gamma -1}\bigr)- \bigl\vert H'(\rho ) (\rho _{ \varepsilon }-\rho ) \bigr\vert - \bigl\vert H(\rho ) \bigr\vert \\ & \geqslant c \rho _{\varepsilon }^{\gamma }-c \rho _{\varepsilon }-c \vert \rho _{\varepsilon }-\rho \vert -c, \end{aligned} $$

that is,

$$ \mathcal{H}(\rho _{\varepsilon })+c \rho _{\varepsilon }+c \vert \rho _{ \varepsilon }-\rho \vert +c\geqslant c \rho _{\varepsilon }^{\gamma }. $$

(3.1)

According to Lemma 5.1 in [7] there exists a constant $c>0$ such that

$$ \begin{aligned}[b] \mathcal{H}(\rho _{\varepsilon }) & \geqslant \inf_{\xi \in \partial \mathcal{O}_{\mathrm{ess}}} \bigl\{ H(\xi )-H'(\rho ) (\xi -\rho )-H( \rho ) \bigr\} \\ &=c(\underline{\rho }, \overline{\rho })>0,\quad \mbox{for all }\rho _{ \varepsilon }\in \mathcal{O}_{\mathrm{res}}. \end{aligned} $$

(3.2)

On the other hand, we know from Lemma 3.1 that

$$ \mathcal{H}(\rho _{\varepsilon })\geqslant c \vert \rho _{\varepsilon }- \rho \vert =c(\rho -\rho _{\varepsilon })\geqslant c\rho _{\varepsilon }, \quad \mbox{for } \rho _{\varepsilon }< \underline{\rho } \leqslant \frac {1}{2}\rho , $$

which together with (3.2) gives

$$ \mathcal{H}(\rho _{\varepsilon })\geqslant c \vert \rho _{\varepsilon }- \rho \vert +c\rho _{\varepsilon }+c, \quad \mbox{for } \rho _{ \varepsilon }< \underline{\rho }\leqslant \frac {1}{2}\rho . $$

Similarly to the above inequality, one has

$$ \mathcal{H}(\rho _{\varepsilon })\geqslant c\rho _{\varepsilon } \geqslant c(\rho _{\varepsilon }-\rho )=c \vert \rho _{\varepsilon }-\rho \vert , \quad \mbox{for } \rho _{\varepsilon }>\overline{\rho }>\rho , $$

then

$$ \mathcal{H}(\rho _{\varepsilon })\geqslant c\rho _{\varepsilon }+c \vert \rho _{\varepsilon }-\rho \vert +c,\quad \mbox{for } \rho _{ \varepsilon }> \overline{\rho }>\rho . $$

In conclusion, we obtain

$$ \mathcal{H}(\rho _{\varepsilon })\geqslant c \vert \rho _{\varepsilon }- \rho \vert +c\rho _{\varepsilon }+c,\quad \mbox{for } \rho _{ \varepsilon }\in \mathcal{O}_{\mathrm{res}}, $$

which together with (3.1) and (3.2) completes the proof. □

Lemma 3.3

Let$\varOmega _{\varepsilon }=(0, \varepsilon )\times (0, 1)$and$u_{\varepsilon }\in W^{1, 2}(\varOmega _{\varepsilon })$be such that$u_{\varepsilon }\cdot n=0$on$\partial \varOmega _{\varepsilon }$. Then there exists some positive constantcindependent ofεsuch that

$$ c \int _{\varOmega _{\varepsilon }} \vert \nabla _{x}u_{\varepsilon } \vert ^{2} \,dx \leqslant \int _{\varOmega _{\varepsilon }}\mathbb{S}(\nabla _{x}u_{ \varepsilon }): \nabla _{x}u_{\varepsilon } \,dx. $$

Proof

By straightforward calculation, we can get

$$ \begin{aligned}[b] & \int _{\varOmega _{\varepsilon }} \bigl(\mu \bigl(\nabla _{x}u_{ \varepsilon }+ \nabla _{x}^{t} u_{\varepsilon }\bigr)+\lambda (\rho _{\varepsilon })\operatorname{div}_{x}u_{\varepsilon }\mathbb{I} \bigr): \nabla _{x}u_{ \varepsilon }\,dx \\ &\quad = \int _{\varOmega _{\varepsilon }}\mu \bigl(\partial _{i}u_{\varepsilon }^{j}+ \partial _{j}u_{\varepsilon }^{i}\bigr)\partial _{i}u_{\varepsilon }^{j}+ \lambda (\rho _{\varepsilon }) \bigl(\operatorname{div}_{x}u_{\varepsilon } \partial _{1}u_{\varepsilon }^{1}+\operatorname{div}_{x}u_{\varepsilon } \partial _{2}u_{\varepsilon }^{2}\bigr)\,dx \\ &\quad = \int _{\varOmega _{\varepsilon }} \bigl(\mu \vert \nabla _{x}u_{\varepsilon } \vert ^{2}+ \lambda (\rho _{\varepsilon }) \vert \operatorname{div}_{x}u_{\varepsilon } \vert ^{2} \bigr)\,dx+ \mu \int _{\varOmega _{\varepsilon }}\partial _{j}u_{\varepsilon }^{i} \partial _{i}u_{\varepsilon }^{j} \,dx. \end{aligned} $$

(3.3)

Now, we estimate the last term of the above equality. According to the boundary condition (1.2), one obtains

$$ \textstyle\begin{cases} u^{1}=\partial _{1}u^{2}=0 \quad \mbox{on } \{x_{1}=0, \varepsilon \}, \\ u^{2}=\partial _{2}u^{1}=0 \quad \mbox{on } \{ x_{2}=0, 1\}. \end{cases} $$

(3.4)

Thus, using (3.4) and integrating by parts on $\varOmega _{\varepsilon }$, we have

$$ \begin{aligned} & \int _{\varOmega _{\varepsilon }}\partial _{j}u_{\varepsilon }^{i} \partial _{i}u_{\varepsilon }^{j} \,dx \\ &\quad = \int _{0}^{1} \int _{0}^{\varepsilon }\partial _{1}u_{\varepsilon }^{i} \partial _{i}u_{\varepsilon }^{1} \,dx_{1} \,dx_{2}+ \int _{0}^{\varepsilon } \int _{0}^{1} \partial _{2}u_{\varepsilon }^{i} \partial _{i}u_{\varepsilon }^{2} \,dx_{2} \,dx_{1} \\ &\quad =- \int _{0}^{1} \int _{0}^{\varepsilon }u_{\varepsilon }^{i} \partial _{i} \partial _{1}u_{\varepsilon }^{1} \,dx_{1}\,dx_{2}- \int _{0}^{\varepsilon } \int _{0}^{1} u_{\varepsilon }^{i} \partial _{i} \partial _{2} u_{\varepsilon }^{2} \,dx_{2}\,dx_{1} \\ &\quad = \int _{0}^{1} \int _{0}^{\varepsilon }\partial _{i}u_{\varepsilon }^{i} \partial _{1}u_{\varepsilon }^{1} \,dx_{1} \,dx_{2}+ \int _{0}^{\varepsilon } \int _{0}^{1} \partial _{i}u_{\varepsilon }^{i} \partial _{2} u_{\varepsilon }^{2} \,dx_{2} \,dx_{1} \\ &\quad = \int _{\varOmega _{\varepsilon }} \vert \operatorname{div}_{x}u_{\varepsilon } \vert ^{2}\,dx, \end{aligned} $$

which together with (3.3) yields

$$ \begin{aligned}[b] & \int _{\varOmega _{\varepsilon }} \bigl(\mu \bigl(\nabla _{x}u_{ \varepsilon }+ \nabla _{x}^{t} u_{\varepsilon }\bigr)+\lambda (\rho _{\varepsilon })\operatorname{div}_{x}u_{\varepsilon }\mathbb{I} \bigr): \nabla _{x}u_{ \varepsilon }\,dx \\ &\quad = \int _{\varOmega _{\varepsilon }} \bigl(\mu \vert \nabla _{x}u_{\varepsilon } \vert ^{2}+\bigl( \lambda (\rho _{\varepsilon })+\mu \bigr) \vert \operatorname{div}_{x}u_{\varepsilon } \vert ^{2} \bigr)\,dx. \end{aligned} $$

(3.5)

By the definition of $\mathbb{S}(\nabla _{x}u_{\varepsilon })$, we get from (3.5)

$$ \begin{aligned}[b] & \int _{\varOmega _{\varepsilon }}\mathbb{S}(\nabla _{x}u_{ \varepsilon }): \nabla _{x}u_{\varepsilon } \,dx \\ &\quad = \int _{\varOmega _{\varepsilon }} \bigl(\mu \vert \nabla _{x}u_{ \varepsilon } \vert ^{2}+\bigl(\lambda (\rho _{\varepsilon })+\mu \bigr) \vert \operatorname{div}_{x}u_{ \varepsilon } \vert ^{2} \bigr)\,dx \\ &\quad \geqslant c \int _{\varOmega _{\varepsilon }} \vert \nabla _{x}u_{ \varepsilon } \vert ^{2} \,dx, \end{aligned} $$

(3.6)

this completes the proof. □

Proof of Theorem 2.1

In order to prove Theorem 2.1, we take

$$ r=\rho (y, t),\qquad U=v(y, t)= \begin{pmatrix} 0 \\ u(y, t) \end{pmatrix}. $$

It follows from Remark 2.1 and (2.3) that

$$ \begin{aligned}[b] &\mathcal{E}_{\varepsilon } \bigl([ \rho _{\varepsilon }, u_{ \varepsilon }]\big| [\rho , v] \bigr) + \frac {1}{ \vert I_{\varepsilon } \vert } \int _{0}^{\tau } \int _{\varOmega _{\varepsilon }}\mathbb{S} \bigl(\nabla _{x}(u_{ \varepsilon }-v) \bigr): \nabla _{x}(u_{\varepsilon }-v)\,dx \,dt \\ &\quad \leqslant \mathcal{E}_{\varepsilon } \bigl([\rho _{0, \varepsilon }, u_{0, \varepsilon }]\big| [\rho _{0}, v_{0}] \bigr)+ \int _{0}^{\tau } \mathcal{F}_{\varepsilon }(\rho _{\varepsilon }, u_{\varepsilon }, \rho , v)\,dt, \end{aligned} $$

(3.7)

where the (scaled) relative entropy functional

$$ \begin{aligned}[b] &\mathcal{E}_{\varepsilon } \bigl([ \rho _{\varepsilon }, u_{ \varepsilon }]\big| [\rho , v] \bigr) \\ &\quad = \frac {1}{ \vert I_{\varepsilon } \vert } \int _{0}^{1} \int _{I_{\varepsilon }} \biggl(\frac {1}{2}\rho _{ \varepsilon } \vert u_{\varepsilon }-v \vert ^{2}+H(\rho _{\varepsilon })-H'( \rho ) (\rho _{\varepsilon }-\rho )-H(\rho ) \biggr)\,dx_{1} \,dx_{2} \\ &\quad = \frac {1}{ \vert I_{\varepsilon } \vert } \int _{0}^{1} \int _{I_{\varepsilon }} \biggl(\frac {1}{2}\rho _{\varepsilon } \vert u_{\varepsilon }-v \vert ^{2}+ \mathcal{H}(\rho _{\varepsilon }) \biggr) \end{aligned} $$

(3.8)

and the remainder $\mathcal{F}_{\varepsilon }$ reads

$$ \begin{aligned}[b] \mathcal{F}_{\varepsilon }(\rho _{\varepsilon }, u_{ \varepsilon }, \rho , v) &=\frac {1}{ \vert I_{\varepsilon } \vert } \int _{\varOmega _{ \varepsilon }}\rho _{\varepsilon }(v_{t}+u_{\varepsilon } \cdot \nabla _{x} v)\cdot (v-u_{\varepsilon })\,dx \\ &\quad {} +\frac {1}{ \vert I_{\varepsilon } \vert } \int _{\varOmega _{\varepsilon }} \rho (v_{t}+v\cdot \nabla _{x} v)\cdot (u_{\varepsilon }-v)\,dx \\ &\quad {} +\frac {1}{ \vert I_{\varepsilon } \vert } \int _{\varOmega _{\varepsilon }} \bigl[P(\rho )-P(\rho _{\varepsilon })+(\rho _{\varepsilon }-\rho )P'( \rho ) \bigr] \operatorname{div}_{x}v \,dx \\ &\quad {} +\frac {1}{ \vert I_{\varepsilon } \vert } \int _{\varOmega _{\varepsilon }} ( \rho _{\varepsilon }-\rho ) (u_{\varepsilon }-v)\cdot \nabla _{x}H'( \rho )\,dx \\ &\triangleq \sum_{i=1}^{3}J_{i}. \end{aligned} $$

(3.9)

In order to handle the integrals on the right-hand side of (3.9), we proceed in several steps as follows.

Step 1. Observe that by (2.6) and (2.7) in Theorem 2.1, we get

$$ \begin{aligned}[b] &\mathcal{E}_{\varepsilon } \bigl([ \rho _{0, \varepsilon }, u_{0, \varepsilon }]\big| [\rho _{0}, v_{0}] \bigr) \\ &\quad =\frac {1}{ \vert I_{\varepsilon } \vert } \int _{\varOmega _{\varepsilon }} \biggl( \frac {1}{2}\rho _{0, \varepsilon } \vert u_{0, \varepsilon }-v_{0} \vert ^{2}+H( \rho _{0, \varepsilon })-H'(\rho _{0}) (\rho _{0, \varepsilon }-\rho _{0})-H( \rho _{0}) \biggr)\,dx \to 0, \end{aligned} $$

(3.10)

as $\varepsilon \to 0$. From now on, we include this term in $\varGamma (\varepsilon )$ where $\varGamma (\varepsilon ) \to 0$ as $\varepsilon \to 0$.

On the other hand, by Lemma 3.3, we obtain

$$ \begin{aligned}[b] & \int _{\varOmega _{\varepsilon }}\mathbb{S} \bigl(\nabla _{x}(u_{ \varepsilon }-v) \bigr): \nabla _{x}(u_{\varepsilon }-v)\,dx \\ &\quad \geqslant c \int _{\varOmega _{\varepsilon }} \bigl\vert \nabla _{x}(u_{ \varepsilon }-v) \bigr\vert ^{2}\,dx \\ &\quad \geqslant c \int _{\varOmega _{\varepsilon }} \bigl\vert \partial _{y} \bigl(u_{ \varepsilon }^{2}-u\bigr) \bigr\vert ^{2} \,dx, \end{aligned} $$

(3.11)

Due to the condition $u_{\varepsilon }\cdot n|_{\partial \varOmega _{\varepsilon }}=0$ and $v\cdot n|_{\partial \varOmega }=0$, we know that $u_{\varepsilon }^{2}=u=0$ on $\{y=0, 1\}$, and then, by the Poincaré inequality, we have

$$ c \int _{\varOmega _{\varepsilon }} \bigl\vert u_{\varepsilon }^{2}-u \bigr\vert ^{2} \,dx \leqslant \int _{\varOmega _{\varepsilon }} \bigl\vert \partial _{y} \bigl(u_{ \varepsilon }^{2}-u\bigr) \bigr\vert ^{2} \,dx, $$

which together with (3.11) gives

$$ \int _{\varOmega _{\varepsilon }}\mathbb{S} \bigl(\nabla _{x}(u_{ \varepsilon }-v) \bigr): \nabla _{x}(u_{\varepsilon }-v)\,dx \geqslant c \int _{\varOmega _{\varepsilon }} \bigl\vert u_{\varepsilon }^{2}-u \bigr\vert ^{2} \,dx. $$

(3.12)

Step 2. By Eq. (1.4)₂, we have

$$\begin{aligned} J_{1} &= \frac {1}{ \vert I_{\varepsilon } \vert } \int _{\varOmega _{ \varepsilon }}\rho _{\varepsilon }(v_{t}+u_{\varepsilon } \cdot \nabla _{x} v)\cdot (v-u_{\varepsilon })\,dx \\ &\quad {} +\frac {1}{ \vert I_{\varepsilon } \vert } \int _{\varOmega _{\varepsilon }} \rho (v_{t}+v\cdot \nabla _{x} v)\cdot (u_{\varepsilon }-v)\,dx \\ &= \frac {1}{ \vert I_{\varepsilon } \vert } \int _{\varOmega _{\varepsilon }} \bigl[( \rho -\rho _{\varepsilon })v_{t}+( \rho v-\rho _{\varepsilon }u_{ \varepsilon })\cdot \nabla _{x}v \bigr]\cdot (u_{\varepsilon }-v)\,dx \\ &= \frac {1}{ \vert I_{\varepsilon } \vert } \int _{\varOmega _{\varepsilon }} \bigl[( \rho -\rho _{\varepsilon })v_{t}+( \rho -\rho _{\varepsilon })v\cdot \nabla _{x}v+\rho _{\varepsilon } (v-u_{\varepsilon })\cdot \nabla _{x}v \bigr] \cdot (u_{\varepsilon }-v)\,dx \\ &= \frac {1}{ \vert I_{\varepsilon } \vert } \int _{\varOmega _{\varepsilon }}(\rho - \rho _{\varepsilon }) (v_{t}+v\cdot \nabla _{x}v) \cdot (u_{ \varepsilon }-v) \,dx+\frac {1}{ \vert I_{\varepsilon } \vert } \int _{\varOmega _{ \varepsilon }} \rho _{\varepsilon } (v-u_{\varepsilon })\cdot \nabla _{x}v \cdot (u_{\varepsilon }-v)\,dx \\ &= \frac {1}{ \vert I_{\varepsilon } \vert } \int _{\varOmega _{\varepsilon }}(\rho - \rho _{\varepsilon }) (u_{t}+u\partial _{y}u) \bigl(u_{\varepsilon }^{2}-u \bigr)\,dx- \frac {1}{ \vert I_{\varepsilon } \vert } \int _{\varOmega _{\varepsilon }} \rho _{ \varepsilon } \bigl(u_{\varepsilon }^{2}-u \bigr)^{2}\partial _{y}u \,dx \\ &= \frac {1}{ \vert I_{\varepsilon } \vert } \int _{\varOmega _{\varepsilon }}\frac { \rho -\rho _{\varepsilon }}{\rho }\bigl(u_{\varepsilon }^{2}-u \bigr) \bigl[ \partial _{y}\bigl(\nu (\rho )\partial _{y}u-P(\rho )\bigr) \bigr]\,dx- \frac {2}{ \vert I_{ \varepsilon } \vert } \int _{\varOmega _{\varepsilon }} \frac {1}{2}\rho _{ \varepsilon } \bigl(u_{\varepsilon }^{2}-u\bigr)^{2}\partial _{y}u \,dx \\ &\leqslant ch_{1}(t) \frac {K_{1}}{ \vert I_{\varepsilon } \vert }+ch_{2}(t) \mathcal{E}_{\varepsilon } \bigl([\rho _{\varepsilon }, u_{\varepsilon }] \big| [\rho , v] \bigr), \end{aligned}$$

(3.13)

where

$$\begin{aligned}& h_{1}(t)\triangleq \bigl\Vert \partial _{y} \bigl(\nu (\rho )\partial _{y}u( \cdot , t)-P(\rho ) (\cdot , t) \bigr) \bigr\Vert _{L^{\infty }(0, 1)}, \\& h_{2}(t)\triangleq \bigl\Vert \partial _{y}u(\cdot , t) \bigr\Vert _{L^{\infty }(0, 1)}, \\& K_{1}\triangleq \int _{\varOmega _{\varepsilon }} \vert \rho _{\varepsilon }- \rho \vert \bigl\vert u_{\varepsilon }^{2}-u \bigr\vert \,dx. \end{aligned}$$

Now we estimate $h_{1}(t)$, $h_{2}(t)$ and $K_{1}$, respectively. Firstly, we split $K_{1}$ into three parts:

$$ \begin{aligned}[b] & \int _{\varOmega _{\varepsilon }} \vert \rho _{\varepsilon }- \rho \vert \bigl\vert u_{\varepsilon }^{2}-u \bigr\vert \,dx \\ &\quad = \int _{\{\underline{\rho }\leqslant \rho _{\varepsilon } \leqslant \overline{\rho }\}} \vert \rho _{\varepsilon }-\rho \vert \bigl\vert u_{ \varepsilon }^{2}-u \bigr\vert \,dx+ \int _{\{\rho _{\varepsilon }< \underline{\rho }\}} \vert \rho _{\varepsilon }-\rho \vert \bigl\vert u_{\varepsilon }^{2}-u \bigr\vert \,dx \\ &\qquad {} + \int _{\{\rho _{\varepsilon }>\overline{\rho }\}} \vert \rho _{ \varepsilon }-\rho \vert \bigl\vert u_{\varepsilon }^{2}-u \bigr\vert \,dx. \end{aligned} $$

(3.14)

Using Lemma 3.1 and Cauchy–Schwarz’s inequality, we obtain

$$\begin{aligned} &\int _{\{\underline{\rho }\leqslant \rho _{\varepsilon } \leqslant \overline{\rho }\}} \vert \rho _{\varepsilon }-\rho \vert \bigl\vert u_{ \varepsilon }^{2}-u \bigr\vert \,dx \\ &\quad \leqslant \delta \int _{\varOmega _{\varepsilon }} \bigl\vert u_{\varepsilon }^{2}-u \bigr\vert ^{2} \,dx+c(\delta ) \int _{\{\underline{\rho }\leqslant \rho _{\varepsilon } \leqslant \overline{\rho }\}} \vert \rho _{\varepsilon }-\rho \vert ^{2}\,dx \\ &\quad \leqslant \delta \int _{\varOmega _{\varepsilon }} \bigl\vert u_{\varepsilon }^{2}-u \bigr\vert ^{2} \,dx+c(\delta ) \int _{\{\underline{\rho }\leqslant \rho _{\varepsilon } \leqslant \overline{\rho }\}}\mathcal{H}(\rho _{\varepsilon })\,dx \\ &\quad \leqslant \delta \int _{\varOmega _{\varepsilon }} \bigl\vert u_{\varepsilon }^{2}-u \bigr\vert ^{2} \,dx+c(\delta ) \vert I_{\varepsilon } \vert \mathcal{E}_{\varepsilon } \bigl([\rho _{ \varepsilon }, u_{\varepsilon }]\big| [ \rho , v] \bigr) \end{aligned}$$

(3.15)

for any $\delta >0$.

Similarly to (3.15), for any $\delta >0$, one has

$$ \begin{aligned}[b] & \int _{\{\rho _{\varepsilon }< \underline{\rho }\}} \vert \rho _{\varepsilon }-\rho \vert \bigl\vert u_{\varepsilon }^{2}-u \bigr\vert \,dx \\ &\quad \leqslant \delta \int _{\varOmega _{\varepsilon }} \bigl\vert u_{\varepsilon }^{2}-u \bigr\vert ^{2} \,dx+c(\delta ) \int _{ \{\rho _{\varepsilon }< \underline{\rho }\}} \vert \rho _{\varepsilon }-\rho \vert ^{2}\,dx \\ &\quad \leqslant \delta \int _{\varOmega _{\varepsilon }} \bigl\vert u_{\varepsilon }^{2}-u \bigr\vert ^{2} \,dx+c(\delta , \underline{\rho }, \overline{\rho }) \int _{ \{\rho _{ \varepsilon }< \underline{\rho }\}} \vert \rho _{\varepsilon }-\rho \vert \,dx \\ &\quad \leqslant \delta \int _{\varOmega _{\varepsilon }} \bigl\vert u_{\varepsilon }^{2}-u \bigr\vert ^{2} \,dx+c(\delta , \underline{\rho }, \overline{\rho } ) \int _{\varOmega _{\varepsilon }}\mathcal{H}(\rho _{\varepsilon })\,dx \\ &\quad \leqslant \delta \int _{\varOmega _{\varepsilon }} \bigl\vert u_{\varepsilon }^{2}-u \bigr\vert ^{2} \,dx+c(\delta , \underline{\rho }, \overline{\rho } ) \vert I_{\varepsilon } \vert \mathcal{E}_{\varepsilon } \bigl([\rho _{\varepsilon }, u_{\varepsilon }] \big| [\rho , v] \bigr). \end{aligned} $$

(3.16)

Finally, it follows from $\rho _{\varepsilon }>\bar{\rho }$ that

$$ \vert \rho _{\varepsilon }-\rho \vert ^{2}=\rho _{\varepsilon }^{2}-2\rho _{\varepsilon }\rho +\rho ^{2}\leqslant \rho _{\varepsilon }^{2}+\rho ^{2} \leqslant \rho _{\varepsilon }^{2}+\bar{\rho }^{2}\leqslant 2\rho _{\varepsilon }^{2}, $$

thus

$$ \begin{aligned}[b] & \int _{\{\rho _{\varepsilon }>\overline{\rho }\}} \vert \rho _{ \varepsilon }-\rho \vert \bigl\vert u_{\varepsilon }^{2}-u \bigr\vert \,dx \\ &\quad \leqslant c \int _{\varOmega _{\varepsilon }}\rho _{\varepsilon } \bigl\vert u_{ \varepsilon }^{2}-u \bigr\vert ^{2} \,dx+c \int _{\{\rho _{\varepsilon }> \overline{\rho }\}} \frac { \vert \rho _{\varepsilon }-\rho \vert ^{2}}{\rho _{ \varepsilon }} \,dx \\ &\quad \leqslant c \int _{\varOmega _{\varepsilon }}\rho _{\varepsilon } \vert u_{ \varepsilon }-v \vert ^{2} \,dx+c \int _{\{\rho _{\varepsilon }> \overline{\rho }\}} \rho _{\varepsilon } \,dx \\ &\quad \leqslant c \vert I_{\varepsilon } \vert \mathcal{E}_{\varepsilon } \bigl([ \rho _{\varepsilon }, u_{\varepsilon }]\big| [\rho , v] \bigr). \end{aligned} $$

(3.17)

Combining with (3.14)–(3.17), we have

$$ K_{1} \leqslant \delta \int _{\varOmega _{\varepsilon }} \bigl\vert u_{\varepsilon }^{2}-u \bigr\vert ^{2} \,dx+c(\delta ) \vert I_{\varepsilon } \vert \mathcal{E}_{\varepsilon } \bigl([\rho _{ \varepsilon }, u_{\varepsilon }]\big| [ \rho , v] \bigr), $$

(3.18)

which together with (3.13), and by the regularity of $(\rho , u)$, leaves us with

$$ J_{1}\leqslant \delta \frac {h_{1}(t)}{ \vert I_{\varepsilon } \vert } \int _{ \varOmega _{\varepsilon }} \bigl\vert u_{\varepsilon }^{2}-u \bigr\vert ^{2} \,dx+c(\delta ) \bigl[h_{1}(t)+h_{2}(t) \bigr] \mathcal{E}_{\varepsilon } \bigl([\rho _{ \varepsilon }, u_{\varepsilon }]\big| [\rho , v] \bigr), $$

(3.19)

for any $\delta >0$.

Step 3. We split $J_{2}$ into the residual and essential parts:

$$ \begin{aligned}[b] J_{2} &= \frac {1}{ \vert I_{\varepsilon } \vert } \int _{\varOmega _{ \varepsilon }} \bigl[P(\rho )-P(\rho _{\varepsilon })+(\rho _{ \varepsilon }-\rho )P'(\rho ) \bigr] \operatorname{div}_{x}v \,dx \\ & \leqslant \Vert \partial _{y}u \Vert _{L^{\infty }(0, 1)} \frac {1}{ \vert I_{ \varepsilon } \vert } \int _{\varOmega _{\varepsilon }} \bigl\vert \bigl[P(\rho )-P( \rho _{\varepsilon })+(\rho _{\varepsilon }-\rho )P'(\rho ) \bigr]_{\mathrm{ess}} \bigr\vert \,dx \\ &\quad {} + \Vert \partial _{y}u \Vert _{L^{\infty }(0, 1)} \frac {1}{ \vert I_{ \varepsilon } \vert } \int _{\varOmega _{\varepsilon }} \bigl\vert \bigl[P(\rho )-P( \rho _{\varepsilon })+(\rho _{\varepsilon }-\rho )P'(\rho ) \bigr]_{\mathrm{res}} \bigr\vert \,dx \\ &= \frac {h_{2}(t)}{ \vert I_{\varepsilon } \vert }(K_{2}+K_{3}), \end{aligned} $$

(3.20)

where

$$\begin{aligned}& K_{2}\triangleq \int _{\varOmega _{\varepsilon }} \bigl\vert \bigl[P(\rho )-P( \rho _{\varepsilon })+(\rho _{\varepsilon }-\rho )P'(\rho ) \bigr]_{\mathrm{ess}} \bigr\vert \,dx, \\& K_{3}\triangleq \int _{\varOmega _{\varepsilon }} \bigl\vert \bigl[P(\rho )-P( \rho _{\varepsilon })+(\rho _{\varepsilon }-\rho )P'(\rho ) \bigr]_{\mathrm{res}} \bigr\vert \,dx. \end{aligned}$$

We immediately see from Lemma 3.1 and the Taylor theorem that

$$ \begin{aligned}[b] K_{2} & \leqslant c \int _{\varOmega _{\varepsilon }} \bigl[ \vert \rho _{\varepsilon }-\rho \vert ^{2} \bigr]_{\mathrm{ess}} \,dx \\ & \leqslant c \int _{\varOmega _{\varepsilon }} \mathcal{H}(\rho _{ \varepsilon }) \,dx \\ & \leqslant c \vert I_{\varepsilon } \vert \mathcal{E}_{\varepsilon } \bigl([ \rho _{\varepsilon }, u_{\varepsilon }]\big| [\rho , v] \bigr). \end{aligned} $$

(3.21)

Next, we show that similar estimates hold also for the residual parts. Firstly, by Lemma 3.2 and Cauchy–Schwarz’s inequality, for any $\alpha \in [1, \gamma ]$, we have

$$ \begin{aligned} \int _{\varOmega _{\varepsilon }} \bigl\vert \bigl[\rho _{ \varepsilon }^{\alpha }- \rho ^{\alpha } \bigr]_{\mathrm{res}} \bigr\vert \,dx & \leqslant c \int _{\varOmega _{\varepsilon }} \bigl\vert [\rho _{ \varepsilon } ]_{\mathrm{res}} \bigr\vert ^{\alpha } +1_{\mathrm{res}}\,dx \\ & \leqslant c \int _{\varOmega _{\varepsilon }} \bigl\vert [\rho _{ \varepsilon } ]_{\mathrm{res}} \bigr\vert ^{\gamma } \,dx+c \int _{\varOmega _{ \varepsilon }} 1_{\mathrm{res}}\,dx \\ & \leqslant c \int _{\varOmega _{\varepsilon }}\mathcal{H}(\rho _{ \varepsilon }) \,dx \\ & \leqslant c \vert I_{\varepsilon } \vert \mathcal{E}_{\varepsilon } \bigl([ \rho _{\varepsilon }, u_{\varepsilon }]\big| [\rho , v] \bigr), \end{aligned} $$

thus

$$ \begin{aligned}[b] K_{3} & \leqslant c \int _{\varOmega _{\varepsilon }} \bigl\vert \bigl[\rho _{\varepsilon }^{\gamma }- \rho ^{\gamma } \bigr]_{\mathrm{res}} \bigr\vert \,dx+c \int _{\varOmega _{\varepsilon }} \bigl\vert [\rho _{\varepsilon }-\rho ]_{\mathrm{res}} \bigr\vert \,dx \\ & \leqslant c \vert I_{\varepsilon } \vert \mathcal{E}_{\varepsilon } \bigl([ \rho _{\varepsilon }, u_{\varepsilon }]\big| [\rho , v] \bigr). \end{aligned} $$

(3.22)

Combining with (3.20)–(3.22), one has

$$ J_{2}\leqslant c h_{2}(t) \mathcal{E}_{\varepsilon } \bigl([\rho _{ \varepsilon }, u_{\varepsilon }]\big| [ \rho , v] \bigr). $$

(3.23)

Step 4. It follows from (3.18) that, for any $\delta >0$, we have

$$ \begin{aligned}[b] J_{3} &= \frac {1}{ \vert I_{\varepsilon } \vert } \int _{\varOmega _{ \varepsilon }} (\rho _{\varepsilon }-\rho ) (u_{\varepsilon }-v)\cdot \nabla _{x}H'(\rho )\,dx \\ &\leqslant \frac {c}{ \vert I_{\varepsilon } \vert } \bigl\Vert \partial _{y}H'( \rho ) \bigr\Vert _{L^{\infty }(0, 1)} \int _{\varOmega _{\varepsilon }} \vert \rho _{ \varepsilon }-\rho \vert \bigl\vert u_{\varepsilon }^{2}-u \bigr\vert \,dx \\ &=\frac {c}{ \vert I_{\varepsilon } \vert } \bigl\Vert \partial _{y}H'( \rho ) \bigr\Vert _{L^{ \infty }(0, 1)}K_{1} \\ &\leqslant \delta h_{3}(t) \biggl(\frac {1}{ \vert I_{\varepsilon } \vert } \int _{ \varOmega _{\varepsilon }} \bigl\vert u_{\varepsilon }^{2}-u \bigr\vert ^{2} \,dx \biggr) \\ &\quad {}+c(\delta )h_{3}(t) \mathcal{E}_{\varepsilon } \bigl([\rho _{\varepsilon }, u_{\varepsilon }] \big| [\rho , v] \bigr), \end{aligned} $$

(3.24)

where

$$ h_{3}(t)\triangleq \bigl\Vert \partial _{y}H'( \rho ) \bigr\Vert _{L^{\infty }(0, 1)}. $$

Inserting (3.10), (3.11), (3.19), (3.23) and (3.24) into (3.7) and (3.9), and taking $\delta >0$ suitable small, we have

$$ \begin{aligned}[b] &\mathcal{E}_{\varepsilon } \bigl([ \rho _{\varepsilon }, u_{ \varepsilon }]\big| [\rho , v] \bigr) (\tau ) + \frac {1}{ \vert I_{\varepsilon } \vert } \int _{0}^{\tau } \int _{\varOmega _{\varepsilon }} \bigl\vert u_{\varepsilon }^{2}-u \bigr\vert ^{2}\,dx \,dt \\ &\quad \leqslant \varGamma (\varepsilon )+c \int _{0}^{\tau } \bigl[ h_{1}(t)+h_{2}(t)+h_{3}(t) \bigr]\mathcal{E}_{\varepsilon } \bigl([\rho _{\varepsilon }, u_{ \varepsilon }]\big| [\rho , v] \bigr) \,dt. \end{aligned} $$

In addition, since $(\rho , u)$ is the solution of (1.4)–(1.6), we have

$$ h_{1}(t)+h_{2}(t)+h_{3}(t)\leqslant c. $$

Thus, a straightforward application of Gronwall’s lemma yields

$$ \mathcal{E}_{\varepsilon } \bigl([\rho _{\varepsilon }, u_{\varepsilon }]\big| [\rho , v] \bigr) (\tau )\to 0 \quad \mbox{as } \varepsilon \to 0. $$

(3.25)

On the other hand, By using Lemma 3.1 and Lemma 3.2, we infer that

$$\begin{aligned}& \vert \rho _{\varepsilon }-\rho \vert ^{2} \geqslant c \vert \rho _{\varepsilon }- \rho \vert ^{\gamma },\quad 1< \gamma \leqslant 2, \mbox{ for } \rho _{\varepsilon }\in \mathcal{O}_{\mathrm{ess}}, \end{aligned}$$

(3.26)

$$\begin{aligned}& \vert \rho _{\varepsilon }-\rho \vert ^{2}= \frac {1}{ \vert \rho _{\varepsilon }-\rho \vert ^{ \gamma -2}} \vert \rho _{\varepsilon }-\rho \vert ^{\gamma } \geqslant c \vert \rho _{ \varepsilon }-\rho \vert ^{\gamma },\quad \gamma \geqslant 2, \mbox{ for } \rho _{\varepsilon }\in \mathcal{O}_{\mathrm{ess}}, \end{aligned}$$

(3.27)

and

$$ c\bigl(1+\rho _{\varepsilon }^{\gamma }\bigr)\geqslant c \vert \rho _{\varepsilon }- \rho \vert ^{\gamma }, \quad \gamma >1, \mbox{ for } \rho _{ \varepsilon }\in \mathcal{O}_{\mathrm{res}}. $$

(3.28)

Hence, it follows from (3.8), Lemma 3.1, Lemma 3.2 and (3.26)–(3.28) that

$$ \begin{aligned} \mathcal{E}_{\varepsilon } \bigl([\rho _{\varepsilon }, u_{ \varepsilon }]\big| [\rho , v] \bigr) &\geqslant \frac {c}{ \vert I_{ \varepsilon } \vert } \int _{\varOmega _{\varepsilon }} \bigl( \bigl\vert [\rho _{ \varepsilon }-\rho ]_{\mathrm{ess}} \bigr\vert ^{2}+ \bigl\vert [1+\rho _{\varepsilon }]_{\mathrm{res}} \bigr\vert ^{\gamma } \bigr)\,dx \\ & + \frac {c}{ \vert I_{\varepsilon } \vert } \int _{\varOmega _{\varepsilon }} \rho _{\varepsilon } \vert u_{\varepsilon }-v \vert ^{2}\,dx \\ &\geqslant \frac {c}{ \vert I_{\varepsilon } \vert } \int _{\varOmega _{\varepsilon }} \bigl( \vert \rho _{\varepsilon }-\rho \vert ^{\gamma }+\rho _{\varepsilon } \vert u_{ \varepsilon }-v \vert ^{2} \bigr)\,dx, \end{aligned} $$

which together with (3.25) completes the proof of Theorem 2.1. □