Global Smooth Solutions for 1D Barotropic Navier–Stokes Equations with a Large Class of Degenerate Viscosities

Abstract

We prove the global existence and uniqueness of smooth solutions to the one-dimensional barotropic Navier–Stokes system with degenerate viscosity $\mu (\rho )=\rho ^\alpha$. We establish that the smooth solutions have possibly two different far-fields, and the initial density remains positive globally in time, for the initial data satisfying the same conditions. In addition, our result works for any $\alpha >0$, i.e., for a large class of degenerate viscosities. In particular, our models include the viscous shallow water equations. This extends the result of Constantin et al. (Ann Inst Henri Poincaré Anal Non Linéaire 37:145–180, 2020, Theorem 1.6) (on the case of periodic domain) to the case where smooth solutions connect possibly two different limits at the infinity on the whole space.

1 Introduction

We consider the one-dimensional barotropic Navier–Stokes system in the Eulerian coordinates:

$$\begin{aligned} \begin{aligned} \left\{ \begin{array}{ll} \rho _t + (\rho u)_x =0,\\ (\rho u)_t+(\rho u^2)_x + p(\rho )_x = (\mu (\rho ) u_x)_x, \end{array} \right. \end{aligned} \end{aligned}$$

(1.1)

where the pressure $p(\rho )$ follows the case of a polytropic perfect gas, i.e.,

$$\begin{aligned} p(\rho )= \rho ^{\gamma },\quad \gamma > 1, \end{aligned}$$

(1.2)

with $\gamma$ the adiabatic constant. Here, $\mu$ denotes the viscosity coefficient given by

$$\begin{aligned} \mu (\rho ) = \rho ^{\alpha }. \end{aligned}$$

(1.3)

Notice that if $\alpha >0$, $\mu (\rho )$ degenerates near the vacuum, i.e., near $\rho =0$. Very often, the viscosity coefficient is assumed to be constant, i.e., $\alpha =0$. However, in the physical context the viscosity of a gas depends on the temperature (see Chapman and Cowling 1970). In the barotropic case, the viscosity depends directly on the density. In general, the viscosity is expected to degenerate on the vacuum as a power of the density as in (1.3).

There are many results on the existence of solutions to the compressible Navier–Stokes equations with the constant viscosity for the one-dimensional case. The existence of weak solutions was first established by Kazhikhov and Shelukhin (1977) for smooth enough initial data close to the equilibrium bounded away from zero. The case of discontinuous data but still bounded away from zero was addressed by Shelukhin (1982, 1983, 1984) and then by Serre (1986) and Hoff (1987a). First result for vanishing initial density was obtained by Shelukhin (1986). Hoff (1987b) proved the existence of global weak solutions with large discontinuous initial data, possibly having different limits at the infinity. There, he also proved that the vacuum cannot form in finite time. The issues on regularity and uniqueness of solutions were first studied by Solonnikov (1976) for smooth initial data and for small time. However, the regularity may blow-up as the solution gets close to vacuum. Hoff and Smoller (2001) show that any weak solution of the one-dimensional Navier–Stokes equations does not have vacuum states for every time, provided that no vacuum states initially exist.

Concerning the 1D existence theory for the degenerate case (1.1), Mellet and Vasseur (2007/08) proved the global existence and uniqueness of strong solutions with large initial data having possibly different limits at the infinity without no vacuum states in the case of $\alpha <1/2$ and $\gamma >1$. To control the $L^\infty$-norm of $1/\rho$ globally in time, they used the relative entropy inequality based on the Bresch–Desjardins entropy, which was derived in Bresch and Desjardins (2002) for the multi-dimensional Korteweg system of equations (for the case of $\alpha =1$ and with an additional capillary term) and later generalized in Bresch and Desjardins (2004). In the one-dimensional case, a similar inequality was introduced earlier by Vaigant (1990) for flows with constant viscosity.

The result of Mellet and Vasseur (2007/08) was extended by Haspot (2018) to the case of $\alpha \in (1/2,1]$. Recently, Constantin et al. (2020, Theorem 1.6) extended it to the case of $\alpha \ge 0$ and $\gamma \in [\alpha , \alpha +1]$ with $\gamma >1$, but they dealt with it on the periodic domain, and with an additional technical condition [see (1.6)].

In this article, we aim to extend the result (Constantin et al. 2020, Theorem 1.6) to the case where smooth solutions have possibly different limits at the infinity on the whole space. This extended result is motivated by the recent works (Kang and Vasseur 2017, 2019) of the authors on the contraction property, up to a time-dependent shift, for large perturbations of viscous shocks (connecting two different end states at $x=\pm \infty$) for the one-dimensional barotropic Navier–Stokes system with degenerate viscosity. In Kang and Vasseur (2017, 2019), solutions of the Navier–Stokes system need to be regular for existence of the time-dependent shift.

1.1 Main Results

We study global existence of smooth solutions to (1.1) with initial data having possibly two different limits $(\rho _\pm , u_\pm )$ at $x=\pm \infty$, where $\rho _\pm >0$. For that, we let $\bar{\rho }$ and $\bar{u}$ be smooth monotone functions such that

$$\begin{aligned} \bar{\rho } (x)=\rho _\pm >0 \quad \text{ and }\quad \bar{u}(x)=u_\pm ,\quad \text{ when } \pm x\ge 1. \end{aligned}$$

(1.4)

Theorem 1.1

Assume $\gamma>1, \alpha >0,$ and $\gamma \in [\alpha ,\alpha +1]$. Let $\rho _0$ and $u_0$ be the initial data such that

$$\begin{aligned} \begin{array}{lll} \rho _0 -\bar{\rho } \in H^k(\mathbb {R}),&{}\quad u_0 -\bar{u} \in H^k(\mathbb {R}),&{}\quad \text{ for } \text{ some } \text{ integer } k\ge 4,\\ 0<\underline{\kappa }_0 \le \rho _0(x) \le \overline{\kappa }_0,&{}\quad \forall x\in \mathbb {R},&{}\quad \text{ for } \text{ some } \text{ constants } \underline{\kappa }_0, \overline{\kappa }_0, \end{array} \end{aligned}$$

(1.5)

and

$$\begin{aligned} \partial _x u_0(x)\le \rho _0(x)^{\gamma -\alpha },\quad \forall x\in \mathbb {R}, \end{aligned}$$

(1.6)

where $\bar{\rho }$ and $\bar{u}$ are the smooth monotone functions satisfying (1.4).

Then, there exists a global-in-time unique smooth solution $(\rho , u)$ of (1.1)–(1.3) such that for any $T>0$,

$$\begin{aligned}&\rho -\bar{\rho } \in L^\infty (0,T;H^k(\mathbb {R})) \\&u -\bar{u} \in L^\infty (0,T;H^k(\mathbb {R})) \cap L^2(0,T;H^{k+1}(\mathbb {R})). \end{aligned}$$

Moreover, there exists constants $\underline{\kappa }(T)$ and $\overline{\kappa }(T)$ such that

$$\begin{aligned} \underline{\kappa }(T) \le \rho (t,x) \le \overline{\kappa }(T),\quad \forall (t,x)\in [0,T]\times \mathbb {R}. \end{aligned}$$

Remark 1.1

Note that the system (1.1) is equivalent to the one in the mass Lagrangian coordinates for the regularity in Theorem 1.1. Therefore, the above result provides a class of global-in-time solutions smooth enough, in which the authors proved the contraction property (Kang and Vasseur 2017, 2019) for viscous shocks of the barotropic Navier–Stokes system in the mass Lagrangian coordinates, with any large initial data satisfying (1.5) and (1.6).

Remark 1.2

Note from the assumption on $\alpha$ and $\gamma$ that Theorem 1.1 also holds for the viscous shallow water equations (i.e., $\gamma =2$, $\alpha =1$). We refer to Gerbeau and Perthame (2018) for a derivation of the viscous shallow water equations from the incompressible Navier–Stokes equations with free boundary.

Remark 1.3

The initial assumptions on (1.6) and $k\ge 4$ in (1.5) are the same conditions as in Constantin et al. (2020, Theorem 1.5), which is used to control the active potential (2.9) defined by the density and the velocity (see Lemma 2.2).

Remark 1.4

In Kang and Vasseur (2019), the authors showed some stability property of entropy shocks of the Euler system as the inviscid case $\nu =0$ of the Navier–Stokes system:

$$\begin{aligned} \begin{aligned} \left\{ \begin{array}{ll} \rho _t^\nu + (\rho ^\nu u^\nu )_x =0,\\ (\rho ^\nu u^\nu )_t+(\rho ^\nu (u^\nu )^2)_x + p(\rho ^\nu )_x = \nu (\mu (\rho ^\nu ) u^\nu _x)_x.\end{array} \right. \end{aligned} \end{aligned}$$

(1.7)

There, the proof is based on stability for viscous shocks of (1.7), uniform with respect to $\nu$. This theory is to substitute the notion of inviscid limit of the Navier–Stokes system for the notion of entropy solution of the Euler system. More specifically, for any initial data $(\rho ^0, u^0)$ for the inviscid dynamics, consider $\mathcal {F}_{(\rho ^0, u^0)}$ the set of inviscid limits ($\nu \rightarrow 0$) of solutions for (1.7) with suitable initial values $(\rho _0^\nu , u_0^\nu )$ converging to $(\rho ^0, u^0)$. This set can be seen as a generalization of the set of entropy solutions to the Euler system with the initial data $(\rho ^0, u^0)$. In Kang and Vasseur (2019), it was proved that the entropy shocks are stable in this class $\mathcal {F}_{(\rho ^0, u^0)}$. However, the existence of the non-empty class $\mathcal {F}_{(\rho ^0, u^0)}$ is subject to the existence of solutions to the Navier–Stokes system (1.7) for any fixed $\nu >0$. This requirement is achieved by Theorem 1.1. Note that, for the initial value $(\rho _0^\nu , u_0^\nu )$ of (1.7), the technical condition (1.6) corresponds to $\partial _x u_0^\nu (x)\le \nu ^{-1} \rho _0^\nu (x)^{\gamma -\alpha }$, which is not restrictive in the limit process $\nu \rightarrow 0$.

2 Proof of Theorem 1.1

2.1 Idea of Proof

Since we are looking for solutions converging to possibly two different limits $(\rho _\pm , u_\pm )$ at $x=\pm \infty$, we do not expect that solutions are integrable. Thus, as a starting point, we may take advantage of the existence result (Mellet and Vasseur 2007/08), for solutions $(\rho , u)$ to satisfy $\rho -\bar{\rho }, u-\bar{u}\in L^\infty (0,T;L^2(\mathbb {R}))$. However, since the result (Mellet and Vasseur 2007/08) require the assumption $\alpha <1/2$ while we consider any $\alpha >0$, we may perturb the viscosity coefficient (1.3) by adding $\varepsilon \rho ^{1/4}$ with small parameter $\varepsilon$ as in (2.4), under which we ensure the global existence of strong solution $(\rho _\varepsilon ,u_\varepsilon )$ satisfying the $H^1$-spatial regularity and the positive lower bound of the density [see (2.7) and (2.8)].

To remove the $\varepsilon$-dependence of the approximate viscosity $\mu _\varepsilon$ as in (2.21), we may first show that the lower bound of the density $\rho _\varepsilon$ is independent of $\varepsilon$ as in Proposition 2.2. For that, we basically use the idea in Constantin et al. (2020) on the analysis for the time evolution of the active potential (see Lemma 2.2). To perform the analysis, we need at least $H^4$-spatial regularity of $(\rho _\varepsilon ,u_\varepsilon )$, which requires the initial condition (1.5).

2.2 Approximate Viscosity

As mentioned above, we first recall the existence result in Mellet and Vasseur (2007/08) as follows:

Proposition 2.1

(Mellet and Vasseur 2007/08) Let $\rho _0$ and $u_0$ be the initial data such that

$$\begin{aligned} 0<\underline{\kappa }_0 \le \rho _0(x) \le \overline{\kappa }_0, \quad \rho _0 -\bar{\rho } \in H^1(\mathbb {R}),\quad u_0 -\bar{u} \in H^1(\mathbb {R}), \end{aligned}$$

(2.1)

for some constants $\underline{\kappa }_0, \overline{\kappa }_0$. Let $\nu {:}\,\mathbb {R}_+\rightarrow \mathbb {R}_+$ be a function such that for some constants $C>0$ and $q\in [0,1/2)$,

$$\begin{aligned} \begin{aligned} \nu (y)\ge \left\{ \begin{array}{ll} C y^q &{}\quad \forall y\le 1\\ C &{}\quad \forall y\ge 1, \end{array} \right. \end{aligned} \end{aligned}$$

(2.2)

and

$$\begin{aligned} \nu (y)\le C + C y^\gamma \quad \forall y\ge 0. \end{aligned}$$

(2.3)

Then, there exists a global-in-time unique strong solution $(\rho , u)$ of (1.1)–(1.2) with $\mu =\nu$ such that the following holds:

For any $T>0$, there exist positive constants $\underline{\beta }(T)$ and $\overline{\beta }(T)$ such that

$$\begin{aligned}&\rho -\bar{\rho } \in L^\infty (0,T;H^1(\mathbb {R})),\\&u-\bar{u}\in L^\infty (0,T;H^1(\mathbb {R}))\cap L^2(0,T;H^2(\mathbb {R})),\\&\underline{\beta }(T) \le \rho (t,x) \le \overline{\beta }(T),\quad \forall (t,x)\in [0,T]\times \mathbb {R}. \end{aligned}$$

To use Proposition 2.1, we consider an approximate viscosity coefficient $\mu _\varepsilon$ defined by perturbing the viscosity $\mu$ in (1.3) as follows: For any $0<\varepsilon <1$,

$$\begin{aligned} \mu _\varepsilon (\rho ) :=\max \left( \mu (\rho ), \varepsilon \rho ^{\alpha _*}\right) ,\quad \forall \rho \ge 0,\quad \text{ where } \alpha _*:=\frac{1}{2}\min \left( \alpha ,\frac{1}{2} \right) . \end{aligned}$$

(2.4)

Since

$$\begin{aligned} \mu _\varepsilon (\rho ) \ge \left\{ \begin{array}{ll} \varepsilon \rho ^{1/4} &{}\quad \forall \rho \le 1\\ \varepsilon &{}\quad \forall \rho \ge 1, \end{array} \right. \end{aligned}$$

and it follows from $\gamma \ge \alpha$ that

$$\begin{aligned} \mu _\varepsilon (\rho ) \le 1+ \rho ^\gamma \quad \forall \rho \ge 0, \end{aligned}$$

(2.5)

$\mu _\varepsilon$ satisfies the assumptions (2.2) and (2.3). Therefore, for the initial datum $(\rho _0, u_0)$ satisfying (1.5), Proposition 2.1 implies that there exists a global-in-time unique strong solution $(\rho _\varepsilon , u_\varepsilon )$ of (1.1)–(1.2) with $\mu =\mu _\varepsilon$, i.e.,

$$\begin{aligned} \begin{aligned} \left\{ \begin{array}{ll} \partial _t \rho _\varepsilon + \partial _x(\rho _\varepsilon u_\varepsilon ) =0\\ \partial _t (\rho _\varepsilon u_\varepsilon )+ \partial _x (\rho _\varepsilon u_\varepsilon ^2) + \partial _x p(\rho _\varepsilon ) = \partial _x (\mu _\varepsilon (\rho _\varepsilon ) \partial _x u_\varepsilon )\\ (\rho _\varepsilon , u_\varepsilon ) |_{t=0} =(\rho _0, u_0), \end{array} \right. \end{aligned} \end{aligned}$$

(2.6)

such that the following holds: for any $T>0$, there exist positive constants $\underline{\kappa }_\varepsilon (T)$, $\overline{\kappa }_\varepsilon (T)$ and $C=C(T,\varepsilon , \underline{\kappa }_0,\overline{\kappa }_0)$ such that

$$\begin{aligned} \Vert \rho _\varepsilon -\bar{\rho } \Vert _{L^\infty (0,T;H^1(\mathbb {R}))} + \Vert u_\varepsilon -\bar{u} \Vert _{L^\infty (0,T;H^1(\mathbb {R}))} + \Vert u_\varepsilon -\bar{u} \Vert _{L^2(0,T;H^2(\mathbb {R}))} \le C, \end{aligned}$$

(2.7)

and

$$\begin{aligned} \underline{\kappa }_\varepsilon (T) \le \rho _\varepsilon (t,x) \le \overline{\kappa }_\varepsilon (T),\quad \forall (t,x)\in (0,T)\times \mathbb {R}. \end{aligned}$$

(2.8)

2.3 Higher Sobolev Regularity

For the system (2.6), we consider the active potential

$$\begin{aligned} w_\varepsilon := -p(\rho _\varepsilon ) + \mu _\varepsilon (\rho _\varepsilon ) \partial _x u_\varepsilon . \end{aligned}$$

(2.9)

This is the potential in the momentum equation of (2.6). Indeed, its gradient is the force:

$$\begin{aligned} \rho _\varepsilon ( \partial _tu_\varepsilon +u_\varepsilon \partial _xu_\varepsilon ) = \partial _xw_\varepsilon . \end{aligned}$$

Then, it follows from Constantin et al. (2020, Proposition 3.1) that $w_\varepsilon$ satisfies a forced quadratic heat equation with linear drift:

$$\begin{aligned} \partial _t w_\varepsilon&= \frac{\mu _\varepsilon (\rho _\varepsilon )}{\rho _\varepsilon } \partial _x^2 w_\varepsilon -\left( u_\varepsilon +\mu _\varepsilon (\rho _\varepsilon )\frac{\partial _x\rho _\varepsilon }{\rho _\varepsilon ^2}\right) \partial _xw_\varepsilon \nonumber \\&\quad +\,\left( \rho _\varepsilon \frac{p'(\rho _\varepsilon )}{\mu _\varepsilon (\rho _\varepsilon )} -2p(\rho _\varepsilon ) \frac{\rho _\varepsilon \mu _\varepsilon '(\rho _\varepsilon )+\mu _\varepsilon (\rho _\varepsilon )}{\mu _\varepsilon (\rho _\varepsilon )^2} \right) w_\varepsilon \nonumber \\&\quad -\,\frac{\rho _\varepsilon \mu _\varepsilon '(\rho _\varepsilon )+\mu _\varepsilon (\rho _\varepsilon )}{\mu _\varepsilon (\rho _\varepsilon )^2} w_\varepsilon ^2 + \left( \rho _\varepsilon \frac{p'(\rho _\varepsilon )}{\mu _\varepsilon (\rho _\varepsilon )} -p(\rho _\varepsilon )\frac{\rho _\varepsilon \mu _\varepsilon '(\rho _\varepsilon )+\mu _\varepsilon (\rho _\varepsilon )}{\mu _\varepsilon (\rho _\varepsilon )^2} \right) p(\rho _\varepsilon ). \end{aligned}$$

(2.10)

Note that the new viscosity coefficient $\mu _\varepsilon (\rho _\varepsilon )/\rho _\varepsilon$ of the parabolic Eq. (2.10) on $w_\varepsilon$ is less degenerate than the viscosity coefficient $\mu _\varepsilon (\rho _\varepsilon )$ of the momentum equation in (2.6). Through the coupled system of (2.10) and the continuity equation (2.6)$_1$, we obtain the higher Sobolev regularity of $\rho _\varepsilon$ and $w_\varepsilon$ as long as $\rho _\varepsilon$ is positive [that is guaranteed by (2.8)] as follows:

Lemma 2.1

Let $\gamma ,\alpha$ be any real numbers. Assume that the initial data $\rho _0$ and $u_0$ satisfy

$$\begin{aligned} \begin{array}{lll} \rho _0 -\bar{\rho } \in H^k(\mathbb {R}),\quad u_0 -\bar{u} \in H^k(\mathbb {R}),&{}\quad \text{ for } \text{ some } \text{ integer } k\ge 2,\\ 0<\underline{\kappa }_0 \le \rho _0(x) \le \overline{\kappa }_0,&{} \quad \forall x\in \mathbb {R}, \end{array} \end{aligned}$$

(2.11)

for some constants $\underline{\kappa }_0, \overline{\kappa }_0$. Then, there exists a global-in-time unique smooth solution $(\rho _\varepsilon , u_\varepsilon )$ of (2.6) such that the following holds: For any $T>0$, there exists positive constants $\underline{\kappa }_\varepsilon (T)$, $\overline{\kappa }_\varepsilon (T)$ and $C=C(T,\gamma ,\alpha , k,\varepsilon , \underline{\kappa }_0,\overline{\kappa }_0)$ such that (2.7), (2.8) and

$$\begin{aligned}&\Vert \partial _x^k \rho _\varepsilon \Vert _{L^\infty (0,T;L^2(\mathbb {R}))} + \Vert \partial _x^{k-1} w_\varepsilon \Vert _{L^\infty (0,T;L^2(\mathbb {R}))}+\Vert \partial _x^{k} w_\varepsilon \Vert _{L^2(0,T;L^2(\mathbb {R}))} \\&\quad +\,\Vert \partial _x^k u_\varepsilon \Vert _{L^\infty (0,T;L^2(\mathbb {R}))} + \Vert \partial _x^{k+1} u_\varepsilon \Vert _{L^2(0,T;L^2(\mathbb {R}))} \le C. \end{aligned}$$

This follows straightforwardly from Constantin et al. (2020, Lemma 4.2 and 4.3) when $\Vert w_\varepsilon \Vert _{L^\infty (0,T;L^2(\mathbb {R}))}$ is bounded. However, for the density having two different limits at the infinity, we do not have a $L^2$-bound on $w_\varepsilon (t,x)$ for each t. Therefore, we may prove Lemma 2.1 without using a $L^2$-bound on $w_\varepsilon$. Although we need a slight modification of the proof in Constantin et al. (2020), we present details of the proof in “Appendix A” for the sake of completeness and the justification on uniformity of the high Sobolev norms in Proposition 2.4.

2.4 Uniform Lower Bound for the Density

Lemma 2.2

Assume the same hypotheses as in Theorem 1.1. Then, for any $T>0$, there exist positive constants $C_\gamma$ and $\varepsilon _\gamma$ such that

$$\begin{aligned} w_\varepsilon (t,x) \le C_\gamma \varepsilon ^\theta , \quad \forall \varepsilon \le \varepsilon _\gamma , \quad \forall t\le T, \quad \forall x\in \mathbb {R}, \end{aligned}$$

where $\theta$ is the positive constant as follows:

$$\begin{aligned} \theta :=\frac{\gamma }{\alpha -\alpha _*},\quad \text{ where } \alpha _* \text{ is } \text{ the } \text{ constant } \text{ as } \text{ in } (2.4). \end{aligned}$$

(2.12)

Proof

First of all, using Lemma 2.1 with $k\ge 4$, together with (2.6) and (2.9), we have

$$\begin{aligned} \rho _\varepsilon , u_\varepsilon , w_\varepsilon \in C^1([0,T]\times \mathbb {R}). \end{aligned}$$

Then, note from (2.9), (2.4), (1.2), (1.3) and the initial condition (1.6) that

$$\begin{aligned} w_\varepsilon (0,x) =-p(\rho _0) +\max \left( \mu (\rho _0), \varepsilon \rho _0^{\alpha _*}\right) \partial _x u_0 \le -\rho _0^\gamma + \max \left( \rho _0^\alpha , \varepsilon \rho _0^{\alpha _*}\right) \rho _0^{\gamma -\alpha }. \end{aligned}$$

Since, for all $x\in \mathbb {R}$,

$$\begin{aligned} w_\varepsilon (0,x)&\le \left( -\rho _0^\gamma + \rho _0^\alpha \rho _0^{\gamma -\alpha }\right) \mathbf {1}_{\{\rho _0^\alpha >\varepsilon \rho _0^{\alpha _*}\}} + \left( -\rho _0^\gamma + \varepsilon \rho _0^{\alpha _*} \rho _0^{\gamma -\alpha }\right) \mathbf {1}_{\{\rho _0^\alpha \le \varepsilon \rho _0^{\alpha _*}\}}\\&\le \varepsilon \rho _0^{\gamma -(\alpha -\alpha _*)} \mathbf {1}_{\{\rho _0^\alpha \le \varepsilon \rho _0^{\alpha _*}\}} \le \varepsilon ^{\frac{\gamma }{\alpha -\alpha _*}}, \end{aligned}$$

we have

$$\begin{aligned} w_\varepsilon (0,x) \le \varepsilon ^\theta ,\quad \forall x\in \mathbb {R}. \end{aligned}$$

Since $w_\varepsilon \in C([0,T]\times \mathbb {R})$, if there exists a point $(t_0,x_0)\in (0,T]\times \mathbb {R}$ such that $w_\varepsilon (t_0,x_0)>\varepsilon ^\theta$, then there exists $t_1\ge 0$ such that

$$\begin{aligned} \sup _{x\in \mathbb {R}} w_\varepsilon (t,x)\le \varepsilon ^\theta \quad \forall t\in [0,t_1], \end{aligned}$$

(2.13)

and

$$\begin{aligned} \sup _{x\in \mathbb {R}} w_\varepsilon (t,x)> \varepsilon ^\theta \quad \forall t\in (t_1,t_0]. \end{aligned}$$

Let

$$\begin{aligned} t_2:=\sup \left\{ t\in (t_1, T]~|~ \sup _{x\in \mathbb {R}} w_\varepsilon (t,x)> \varepsilon ^\theta \right\} . \end{aligned}$$

Then,

$$\begin{aligned} \sup _{x\in \mathbb {R}} w_\varepsilon (t,x)\ge \varepsilon ^\theta \quad \forall t\in [t_1,t_2]. \end{aligned}$$

Thus, using the fact that for each $t\le T$,

$$\begin{aligned} w_\varepsilon (t,x) \rightarrow -p(\rho _\pm ) \le 0 \quad \text{ as } ~ x\rightarrow \pm \infty , \end{aligned}$$

we can define the function

$$\begin{aligned} w_M(t):=\max _{x\in \mathbb {R}} w_\varepsilon (t,x), \end{aligned}$$

which is Lipschitz continuous, and differentiable almost everywhere on $[t_1,t_2]$ thanks to the regularity $w_\varepsilon \in C^1([0,T]\times \mathbb {R})$. Moreover, for each $t\in [t_1,t_2]$, there exists $x_t$ such that

$$\begin{aligned} w_M(t)=w_\varepsilon (t,x_t). \end{aligned}$$

Then, $w'_M(t) =(\partial _t w_\varepsilon ) (t,x_t)$ for a.e. $t\in (t_1,t_2)$, since

$$\begin{aligned} w'_M(t)&= \lim _{h\rightarrow 0+} \frac{w_\varepsilon (t+h, x_{t+h}) - w_\varepsilon (t,x_t)}{h} \\&\ge \lim _{h\rightarrow 0+} \frac{w_\varepsilon (t+h, x_{t}) - w_\varepsilon (t,x_t)}{h} = \partial _t w_\varepsilon (t,x_t),\\ w'_M(t)&= \lim _{h\rightarrow 0+} \frac{w_\varepsilon (t,x_t)-w_\varepsilon (t-h, x_{t-h})}{h} \\&\le \lim _{h\rightarrow 0+} \frac{w_\varepsilon (t,x_t)-w_\varepsilon (t-h, x_{t})}{h} = \partial _t w_\varepsilon (t,x_t). \end{aligned}$$

Using this together with $\partial _x^2 w_\varepsilon (t,x_t)\le 0$, $\partial _xw_\varepsilon (t,x_t)=0$ and $\rho _\varepsilon \mu _\varepsilon '(\rho _\varepsilon )\ge 0$, we have from (2.10) that

$$\begin{aligned} w'_M (t) \le J_1(t) w_M(t) + J_2(t),\quad t\in (t_1,t_2), \end{aligned}$$

where (putting $\rho _M(t):=\rho _\varepsilon (t,x_t)$)

$$\begin{aligned} J_1(t)&:= \frac{\rho _M^\gamma }{\mu _\varepsilon (\rho _M)^2} \left( \gamma \mu _\varepsilon (\rho _M) -2 \left( \rho _M\mu _\varepsilon '(\rho _M)+\mu _\varepsilon (\rho _M) \right) \right) , \\ J_2(t)&:= \frac{\rho _M^{2\gamma }}{\mu _\varepsilon (\rho _M)^2} \left( \gamma \mu _\varepsilon (\rho _M) - \left( \rho _M\mu _\varepsilon '(\rho _M)+\mu _\varepsilon (\rho _M) \right) \right) . \end{aligned}$$

Since $\gamma \le \alpha +1$, we have

$$\begin{aligned} J_1(t)&= \frac{\rho _M^\gamma }{\mu _\varepsilon (\rho _M)^2} \left( (\gamma {-}2(\alpha +1))\rho _M^\alpha \mathbf {1}_{\{\rho _M^\alpha {>}\varepsilon \rho _M^{\alpha _*}\}} {+} \varepsilon \left( \gamma -2(\alpha _*+1) \right) \rho _M^{\alpha _*} \mathbf {1}_{\{\rho _M^\alpha \le \varepsilon \rho _M^{\alpha _*}\}} \right) \\&\le \frac{\rho _M^\gamma }{\mu _\varepsilon (\rho _M)^2} \varepsilon \left| \gamma - 2(\alpha _*+1)\right| \rho _M^{\alpha _*} \mathbf {1}_{\{\rho _M^\alpha \le \varepsilon \rho _M^{\alpha _*}\}}. \end{aligned}$$

Moreover, using $\mu _\varepsilon (\rho _M)\ge \varepsilon \rho _M^{\alpha _*}$ and $\mu _\varepsilon (\rho _M)\ge \rho _M^\alpha$ by the definition, we have

$$\begin{aligned} J_1(t) \le \left| \gamma -2(\alpha _*+1) \right| \rho _M^{\gamma -\alpha } \mathbf {1}_{\{\rho _M^\alpha \le \varepsilon \rho _M^{\alpha _*}\}} \le \left| \gamma -2(\alpha _*+1) \right| \varepsilon ^{\frac{\gamma -\alpha }{\alpha -\alpha _*}}. \end{aligned}$$

Likewise, we have

$$\begin{aligned} J_2(t)&= \frac{\rho _M^{2\gamma }}{\mu _\varepsilon (\rho _M)^2} \left( (\gamma -(\alpha +1))\rho _M^\alpha \mathbf {1}_{\{\rho _M^\alpha >\varepsilon \rho _M^{\alpha _*}\}} + \varepsilon \left( \gamma -(\alpha _*+1) \right) \rho _M^{\alpha _*} \mathbf {1}_{\{\rho _M^\alpha \le \varepsilon \rho _M^{\alpha _*}\}} \right) \\&\le \frac{\rho _M^{2\gamma }}{\mu _\varepsilon (\rho _M)^2} \varepsilon \left| \gamma -(\alpha _*+1) \right| \rho _M^{\alpha _*} \mathbf {1}_{\{\rho _M^\alpha \le \varepsilon \rho _M^{\alpha _*}\}}\\&\le \left| \gamma - (\alpha _*+1) \right| \varepsilon ^{\frac{2\gamma -\alpha }{\alpha -\alpha _*}}. \end{aligned}$$

The above estimates and (2.13) imply that for any $t\in [t_1,t_2]$ and $\varepsilon \in (0,1)$,

$$\begin{aligned} w_M(t)&\le w_M(t_1) \exp \left( \int _{t_1}^t J_1(s)ds \right) + \int _{t_1}^t J_2(s) \exp \left( \int _{s}^t J_1(\tau ) d\tau \right) ds\nonumber \\&\le \exp \left( T\left| \gamma - 2(\alpha _*+1) \right| \right) \left( \varepsilon ^\theta + \varepsilon ^{\frac{2\gamma -\alpha }{\alpha -\alpha _*}} T \left| \gamma - (\alpha _*+1) \right| \right) , \end{aligned}$$

(2.14)

If $\gamma >\alpha$, it follows from (2.14) that for all $\varepsilon$ satisfying

$$\begin{aligned} \varepsilon \le \left( \frac{1}{1+ T \left| \gamma -(\alpha _*+1) \right| }\right) ^{\frac{\alpha -\alpha _*}{\gamma -\alpha }}, \end{aligned}$$

the following holds:

$$\begin{aligned} w_M(t) \le 2\exp \left( T\left| \gamma - 2(\alpha _*+1) \right| \right) \varepsilon ^\theta , \quad \forall t\in [t_1,t_2]. \end{aligned}$$

If $\gamma =\alpha$, since $\theta =\frac{2\gamma -\alpha }{\alpha -\alpha _*}$, it follows from (2.14) that

$$\begin{aligned}&w_M(t) \le 2 \left( 1+ T \left| \gamma - (\alpha _*+1) \right| \right) \exp \left( T\left| \gamma - 2(\alpha _*+1) \right| \right) \varepsilon ^\theta ,\nonumber \\&\quad \forall \varepsilon \le 1, \quad \forall t\in [t_1,t_2]. \end{aligned}$$

Therefore, the above estimates together with (2.13) yield that

$$\begin{aligned} \sup _{x\in \mathbb {R}} w_\varepsilon (t,x) \le C_\gamma \varepsilon ^\theta , \quad \forall \varepsilon \le \varepsilon _\gamma , \quad \forall t\in [0,t_2], \end{aligned}$$

where $C_\gamma$ is the constants as in (2.12).

If $t_2 <T$, then the definition of $t_2$ implies

$$\begin{aligned} \sup _{x\in \mathbb {R}} w_\varepsilon (t,x) \le \varepsilon ^\theta , \quad \forall t\in (t_2, T]. \end{aligned}$$

Hence, we complete the proof. $\square$

Proposition 2.2

Assume the same hypotheses as in Theorem 1.1. Then, for any $T>0$, there exist positive constants $\underline{\kappa }(T)=\underline{\kappa }(T)(\gamma , \alpha , \underline{\kappa }_0)$ and $\delta _1=\delta _1(T,\gamma , \alpha , \underline{\kappa }_0)$ (independent of $\varepsilon )$ such that

$$\begin{aligned} \rho _\varepsilon (t,x) \ge \underline{\kappa }(T),\quad \forall t\le T,\quad \forall x\in \mathbb {R}, \quad \forall \varepsilon \le \delta _1. \end{aligned}$$

Proof

Let

$$\begin{aligned} q(\gamma ):=\left\{ \begin{array}{ll} \theta &{}\quad \text{ if } \gamma >\alpha , \\ 1 &{}\quad \text{ if } \gamma =\alpha , \end{array} \right. \quad \text{ where }~ \theta =\frac{\gamma }{\alpha -\alpha _*} \text{ as } \text{ in } \text{ Lemma } 2.2. \end{aligned}$$

We first choose a constant $\delta _1>0$ such that

$$\begin{aligned} \delta _1 := \left\{ \begin{array}{ll} \min \left( \varepsilon _\gamma , \left( \frac{\underline{\kappa }_0}{4} \right) ^{\alpha -\alpha _* }, \left( \frac{2^\alpha -1}{ \alpha (2^\gamma +C_\gamma )T} \right) ^{\frac{\gamma }{q(\gamma )(\gamma -\alpha )}} \right) &{}\quad \text{ if } \gamma >\alpha , \\ \min \left( \varepsilon _\gamma , \left( \frac{\underline{\kappa }_0}{4} \right) ^\alpha , \left( C_\gamma ^{-1}(2^\alpha -1) e^{-\alpha T} \right) ^{\frac{\alpha -\alpha _*}{\alpha _*}}\right) &{}\quad \text{ if } \gamma =\alpha , \end{array} \right. \end{aligned}$$

(2.15)

where $\underline{\kappa }_0$ is the constant as in (1.5), and $\varepsilon _\gamma , C_\gamma$ are the constants as in Lemma 2.2.

Then, since

$$\begin{aligned} \delta _1 \le \left\{ \begin{array}{ll} \left( \frac{\underline{\kappa }_0}{4} \right) ^{\alpha -\alpha _* }&{}\quad \text{ if } \gamma >\alpha , \\ \left( \frac{\underline{\kappa }_0}{4} \right) ^\alpha &{}\quad \text{ if } \gamma =\alpha , \end{array} \right. \end{aligned}$$

we have $2\delta _1^{q(\gamma )/\gamma } < \underline{\kappa }_0$ for any $\gamma \ge \alpha$.

Therefore, it follows from the initial condition of (1.5) that

$$\begin{aligned} \inf _{x\in \mathbb {R}} \rho _0 (x)\ge 2\delta _1^{q(\gamma )/\gamma }. \end{aligned}$$

For any fixed $\varepsilon \le \delta _1$, since $\rho _\varepsilon \in C([0,T]\times \mathbb {R})$, if there exists a point $(t_0,x_0)\in (0,T]\times \mathbb {R}$ such that $\rho _\varepsilon (t_0,x_0)<2\delta _1^{q(\gamma )/\gamma }$, then there exists $t_1\ge 0$ such that

$$\begin{aligned} \inf _{x\in \mathbb {R}} \rho _\varepsilon (t,x)\ge & {} 2\delta _1^{q(\gamma )/\gamma }\quad \forall t\in [0,t_1], \nonumber \\ \inf _{x\in \mathbb {R}} \rho _\varepsilon (t,x)< & {} 2 \delta _1^{q(\gamma )/\gamma }\quad \forall t\in (t_1,t_0]. \end{aligned}$$

(2.16)

Then,

$$\begin{aligned} \inf _{x\in \mathbb {R}} \rho _\varepsilon (t,x)\le 2 \delta _1^{q(\gamma )/\gamma } \quad \forall t\in [t_1,t_2], \end{aligned}$$

(2.17)

where

$$\begin{aligned} t_2:=\sup \left\{ t\in (t_1, T]~|~ \inf _{x\in \mathbb {R}} \rho _\varepsilon (t,x)< 2\delta _1^{q(\gamma )/\gamma } \right\} . \end{aligned}$$

Thus, using $2 \delta _1^{q(\gamma )/\gamma } < \underline{\kappa }_0 \le \min (\rho _-,\rho _+)$ together with the fact that for each $t\le T$,

$$\begin{aligned} \rho _\varepsilon (t,x) \rightarrow \rho _\pm \quad \text{ as } ~ x\rightarrow \pm \infty , \end{aligned}$$

we define the function

$$\begin{aligned} \rho _m(t):=\min _{x\in \mathbb {R}} \rho _\varepsilon (t,x), \end{aligned}$$

which is Lipschitz continuous, and differentiable almost everywhere on $[t_1,t_2]$ thanks to the regularity $\rho _\varepsilon \in C^1([0,T]\times \mathbb {R})$. So, let $y_t$ be a minimizer for $\rho _m(t)=\rho _\varepsilon (t,y_t)$. Since $\rho '_m(t) =(\partial _t \rho _\varepsilon ) (t,y_t)$ for a.e. $t\in (t_1,t_2)$, and $\partial _x\rho _\varepsilon (t,y_t)=0$, we have from the continuity equation of (2.6) that

$$\begin{aligned} \rho '_m(t) = -\rho _m(t) \partial _xu_\varepsilon (y_t),\quad t\in (t_1,t_2). \end{aligned}$$

Then, using (2.9), Lemma 2.2 with $\varepsilon \le \delta _1\le \varepsilon _\gamma$, and $\mu _\varepsilon (\rho _m)\ge \rho _m^\alpha$, we have

$$\begin{aligned} \rho '_m(t) = -\rho _m(t) \frac{p(\rho _m)+w_\varepsilon (y_t)}{\mu _\varepsilon (\rho _m)} \ge -\rho _m^{1+\gamma -\alpha } - C_\gamma \delta _1^\theta \rho _m^{1-\alpha },\quad t\in (t_1,t_2). \end{aligned}$$

(2.18)

Case of$\gamma >\alpha$) Using (2.17) together with $q(\gamma )=\theta$, we have

$$\begin{aligned} \rho '_m \ge - (2^\gamma +C_\gamma ) \delta _1^\theta \rho _m^{1-\alpha }, \end{aligned}$$

which yields

$$\begin{aligned} (\rho _m^\alpha )' \ge - \alpha (2^\gamma +C_\gamma ) \delta _1^\theta ,\quad t\in (t_1,t_2). \end{aligned}$$

Thus, using (2.16), we have

$$\begin{aligned} \rho _m^\alpha (t) \ge \rho _m^\alpha (t_1) - \alpha (2^\gamma +C_\gamma ) \delta _1^\theta T\ge \left( 2\delta _1^{q(\gamma )/\gamma } \right) ^\alpha - \alpha (2^\gamma +C_\gamma )\delta _1^\theta T,\quad \forall t\in [t_1,t_2]. \end{aligned}$$

Since $q(\gamma )=\theta$ when $\gamma >\alpha$, and

$$\begin{aligned} \delta _1 \le \left( \frac{2^\alpha -1}{ \alpha (2^\gamma +C_\gamma )T} \right) ^{\frac{\gamma }{q(\gamma )(\gamma -\alpha )}}, \end{aligned}$$

we have

$$\begin{aligned} \rho _m^\alpha (t) \ge \left( \delta _1^{q(\gamma )/\gamma } \right) ^\alpha ,\quad \forall t\in [t_1,t_2]. \end{aligned}$$

Therefore, this together with (2.16) and the definition of $t_2$ implies

$$\begin{aligned} \inf _{x\in \mathbb {R}} \rho _\varepsilon (t,x)\ge \delta _1^{q(\gamma )/\gamma }\quad \forall t\in [0,T]. \end{aligned}$$

Case of$\gamma =\alpha$ First, it follows from (2.18) with $\gamma =\alpha$ that

$$\begin{aligned} \rho '_m\ge -\rho _m - C_\gamma \delta _1^\theta \rho _m^{1-\alpha } ,\quad t\in (t_1,t_2). \end{aligned}$$

Then, since

$$\begin{aligned} (\rho _m^\alpha )' \ge -\alpha \rho _m^\alpha - \alpha C_\gamma \delta _1^\theta ,\quad t\in (t_1,t_2), \end{aligned}$$

we have

$$\begin{aligned} \rho _m^\alpha (t) \ge \rho _m^\alpha (t_1) e^{-\alpha (t-t_1)} -\alpha C_\gamma \delta _1^\theta \int _{t_1}^t e^{-\alpha (t-s)} \mathrm{d}s, \end{aligned}$$

which together with (2.16) yields

$$\begin{aligned} \rho _m^\alpha (t) \ge \left( 2\delta _1^{q(\gamma )/\gamma } \right) ^\alpha e^{-\alpha T} - C_\gamma \delta _1^\theta ,\quad \forall t\in [t_1,t_2]. \end{aligned}$$

Since $q(\gamma )/\gamma = 1/\alpha$ and $\theta =\alpha /(\alpha -\alpha _*)$ when $\gamma =\alpha$, if needed, taking $\delta _1$ again such that

$$\begin{aligned} \delta _1 \le \left( C_\gamma ^{-1}(2^\alpha -1) e^{-\alpha T} \right) ^{\frac{\alpha -\alpha _*}{\alpha _*}}, \end{aligned}$$

we have

$$\begin{aligned} \rho _m^\alpha (t) \ge e^{-\alpha T} \delta _1,\quad \forall t\in [t_1,t_2]. \end{aligned}$$

Therefore, this together with (2.16) and the definition of $t_2$ implies

$$\begin{aligned} \inf _{x\in \mathbb {R}} \rho _\varepsilon (t,x)\ge e^{-T} \delta _1^{1/\alpha } = e^{-T} \delta _1^{q(\gamma )/\gamma }\quad \forall t\in [0,T]. \end{aligned}$$

Hence, we complete the proof. $\square$

2.5 Uniform Bounds for the Solutions $(\rho _\varepsilon ,u_\varepsilon )$

Thanks to Proposition 2.2, we first have the uniform upper bound for the density as follows:

Proposition 2.3

Under the same hypotheses as in Theorem 1.1, there exists a positive constant $\overline{\kappa }(T)$ (independent of $\varepsilon )$ such that

$$\begin{aligned} \rho _\varepsilon (t,x) \le \overline{\kappa }(T),\quad \forall t\le T,\quad \forall x\in \mathbb {R}, \quad \forall \varepsilon \le \delta _1, \end{aligned}$$

where $\delta _1$ is the constant as in Proposition 2.2.

For the proof of Proposition 2.3, we refer to the proof of Mellet and Vasseur (2007/08, Proposition 4.5), in which the uniform estimates (2.19) and (2.20) are crucially used to get the uniform upper bound $\overline{\kappa }(T)$ of the density: One estimate is on the uniform lower bound of the viscosity $\mu _\varepsilon$ as

$$\begin{aligned} \mu _\varepsilon (\rho _\varepsilon ) \ge \rho _\varepsilon ^\alpha \ge \underline{\kappa }(T)^\alpha ,\quad \forall t\le T,\quad \forall x\in \mathbb {R}, \quad \forall \varepsilon \le \delta _1. \end{aligned}$$

(2.19)

The others are the estimates (Mellet and Vasseur 2007/08, Lemmas 3.1 and 3.2) on the relative entropy related to the Bresch–Desjardins entropy (see Bresch and Desjardins 2002, 2003, 2004) as follows:

$$\begin{aligned}&\sup _{0\le t\le T} \int _\mathbb {R}\left( \rho _\varepsilon \left| u_\varepsilon -\bar{u} \right| ^2 + p(\rho _\varepsilon |\bar{\rho }) \right) \mathrm{d}x + \int _0^T\int _\mathbb {R}\mu _\varepsilon (\rho _\varepsilon ) |\partial _xu_\varepsilon |^2 \mathrm{d}x \mathrm{d}t \le K,\nonumber \\&\sup _{0\le t\le T} \int _\mathbb {R}\left( \rho _\varepsilon \left| ( u_\varepsilon -\bar{u} )+ \partial _x(\varphi (\rho _\varepsilon )) \right| ^2 + p(\rho _\varepsilon |\bar{\rho }) \right) \mathrm{d}x \le K, \end{aligned}$$

(2.20)

where $\varphi '(\rho _\varepsilon ):=\mu _\varepsilon (\rho _\varepsilon )/\rho _\varepsilon ^2$, and the above constant K is independent of $\varepsilon$ thanks to (2.5). Indeed, it follows from Mellet and Vasseur (2007/08, Lemmas 3.1 and 3.2) that the constant K depends only on $T, \gamma , (\bar{\rho }, \bar{u}), (\rho _0,u_0)$, and the constants appearing in (2.3).

Propositions 2.2 and 2.3 together with the above estimates (2.19)–(2.20) imply the following uniform estimates on the Sobolev norms of the solutions $(\rho _\varepsilon ,u_\varepsilon )$:

Proposition 2.4

Under the same hypotheses as in Theorem 1.1, there exists a constant $C\,($independent of $\varepsilon )$ such that

$$\begin{aligned} \Vert \rho _\varepsilon -\bar{\rho } \Vert _{L^\infty (0,T;H^k(\mathbb {R}))} + \Vert u_\varepsilon -\bar{u} \Vert _{L^\infty (0,T;H^k(\mathbb {R}))} + \Vert u_\varepsilon -\bar{u} \Vert _{L^2(0,T;H^{k+1}(\mathbb {R}))} \le C. \end{aligned}$$

For the proof of proposition 2.4, we first refer to the proof of Mellet and Vasseur (2007/08, Proposition 4.6 and 4.7), from which the constant in (2.7) does not depend on $\varepsilon$ anymore. Then, from the proof of Lemma 2.1, we deduce that the constant C in Lemma 2.1 is independent of $\varepsilon$. Therefore, we have Proposition 2.4

2.6 Conclusion

We have shown that for any $\varepsilon \le \delta _1$, the system (2.6) has the unique smooth solution $(\rho _\varepsilon ,u_\varepsilon )$ such that Propositions 2.2–2.4 hold.

We now take $\delta _T$ as

$$\begin{aligned} \delta _T = \min \left( \underline{\kappa }(T)^{\alpha -\alpha _*}, \delta _1 \right) , \end{aligned}$$

where the constants $\underline{\kappa }(T)$ and $\delta _1$ are as in Proposition 2.2.

Then, since Proposition 2.2 implies that for all $\varepsilon <\delta _T$,

$$\begin{aligned} \varepsilon \rho _\varepsilon ^{\alpha _*} < \delta _T \rho _\varepsilon ^{\alpha _*} \le \underline{\kappa }(T)^{\alpha -\alpha _*} \rho _\varepsilon ^{\alpha _*} \le \rho _\varepsilon ^\alpha ,\quad \forall t\le T,\quad \forall x\in \mathbb {R}, \end{aligned}$$

it follows from the definition (1.3) that

$$\begin{aligned} \mu _\varepsilon (\rho _\varepsilon )=\mu (\rho _\varepsilon ), \quad \forall \varepsilon <\delta _T,\quad \forall t\le T,\quad \forall x\in \mathbb {R}. \end{aligned}$$

(2.21)

Recall that the approximate system (2.6) represents the system (1.1) with $\mu _\varepsilon$ instead of $\mu$.

Therefore, for any $T>0$, and any $\varepsilon$ with $\varepsilon <\delta _T$, $(\rho _\varepsilon ,u_\varepsilon )$ is the unique smooth solution of (1.1) with the initial datum $(\rho _0, u_0)$ such that Propositions 2.2–2.4 hold.

Hence, we complete the proof.

Appendix A: Proof of Lemma 2.1

Let $(\rho _\varepsilon , u_\varepsilon )$ be the global strong solution to (2.6) such that (2.7) and (2.8) hold.

Once the desired estimates for $k=2$ are obtained, the remaining part proceeds by induction in k, which follows the same proof of Constantin et al. (2020, Lemma 4.3). Therefore, we here present the proof only when $k=2$, based on the proof of Constantin et al. (2020, Lemma 4.2).

First of all, since $\partial _x u_\varepsilon \in L^2(0,T;L^\infty (\mathbb {R}))$ by (2.7), using (2.7) and (2.8), we have

$$\begin{aligned}&w_\varepsilon \in L^2(0,T;L^\infty (\mathbb {R})), \nonumber \\&\partial _x w_\varepsilon = -p'(\rho _\varepsilon ) \partial _x \rho _\varepsilon +\mu _\varepsilon '(\rho _\varepsilon ) \partial _x \rho _\varepsilon \partial _x u_\varepsilon +\mu _\varepsilon (\rho _\varepsilon ) \partial _x^2 u_\varepsilon \in L^2(0,T;L^2(\mathbb {R})). \end{aligned}$$

(A.1)

Step 1 Differentiating the equation (2.10) in space, multiplying the resulting equation by $\partial _xw_\varepsilon$ and integrating by parts, we have

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} \int _\mathbb {R}\frac{|\partial _xw_\varepsilon |^2}{2} \mathrm{d}x&= -\int _\mathbb {R}\frac{\mu _\varepsilon (\rho _\varepsilon )}{\rho _\varepsilon } |\partial _x^2 w_\varepsilon |^2 \mathrm{d}x +\int _\mathbb {R}\left( u_\varepsilon +\frac{\mu _\varepsilon (\rho _\varepsilon )}{\rho _\varepsilon ^2} \partial _x\rho _\varepsilon \right) \partial _xw_\varepsilon \partial _x^2w_\varepsilon \mathrm{d}x\\&\quad +\,\int _\mathbb {R}f_1(\rho _\varepsilon ) |\partial _xw_\varepsilon |^2 \mathrm{d}x + \int _\mathbb {R}f_1'(\rho _\varepsilon ) \partial _x\rho _\varepsilon w_\varepsilon \partial _xw_\varepsilon \mathrm{d}x \\&\quad -\,2\int _\mathbb {R}f_2(\rho _\varepsilon ) w_\varepsilon |\partial _xw_\varepsilon |^2 \mathrm{d}x \\&\quad -\,\int _\mathbb {R}f_2'(\rho _\varepsilon ) \partial _x\rho _\varepsilon w_\varepsilon ^2 \partial _xw_\varepsilon \mathrm{d}x +\int _\mathbb {R}f_3'(\rho _\varepsilon ) \partial _x\rho _\varepsilon \partial _xw_\varepsilon \mathrm{d}x\\&=: -\int _\mathbb {R}\frac{\mu _\varepsilon (\rho _\varepsilon )}{\rho _\varepsilon } |\partial _x^2 w_\varepsilon |^2 \mathrm{d}x + \sum _{j=1}^6 I_j. \end{aligned}$$

where

$$\begin{aligned} f_1(\rho )&:= \rho \frac{p'(\rho )}{\mu _\varepsilon (\rho )} -2p(\rho ) \frac{\rho \mu _\varepsilon '(\rho )+\mu _\varepsilon (\rho )}{\mu _\varepsilon (\rho )^2},\\ f_2(\rho )&:=\frac{\rho \mu _\varepsilon '(\rho )+\mu _\varepsilon (\rho )}{\mu _\varepsilon (\rho )^2},\\ f_3(\rho )&:=\left( \rho \frac{p'(\rho )}{\mu _\varepsilon (\rho )} -p(\rho )\frac{\rho \mu _\varepsilon '(\rho )+\mu _\varepsilon (\rho )}{\mu _\varepsilon (\rho )^2} \right) p(\rho ). \end{aligned}$$

Since, thanks to (2.8), $L^\infty ([0,T]\times \mathbb {R})$-norms of $\rho _\varepsilon$ to some power are all bounded, there exists a positive constant $C_1=C_1(\underline{\kappa }_\varepsilon (T), \overline{\kappa }_\varepsilon (T))$ such that

$$\begin{aligned} -\int _\mathbb {R}\frac{\mu _\varepsilon (\rho _\varepsilon )}{\rho _\varepsilon } |\partial _x^2 w_\varepsilon |^2 \mathrm{d}x \le - C_1 \int _\mathbb {R}|\partial _x^2 w_\varepsilon |^2 \mathrm{d}x, \end{aligned}$$

and

$$\begin{aligned} \left\| \frac{\mu _\varepsilon (\rho _\varepsilon )}{\rho _\varepsilon ^2} \right\| _{L^\infty ([0,T]\times \mathbb {R})} + \sum _{j=1}^3\left( \Vert f_j(\rho _\varepsilon ) \Vert _{L^\infty ([0,T]\times \mathbb {R})}+ \Vert f_j'(\rho _\varepsilon ) \Vert _{L^\infty ([0,T]\times \mathbb {R})} \right) \le C_1. \end{aligned}$$

Thus, the above terms $I_j$ can be controlled as follows:

$$\begin{aligned} |I_1|&\le \Vert u_\varepsilon \Vert _{L^\infty (\mathbb {R})}\Vert \partial _xw_\varepsilon \Vert _{L^2(\mathbb {R})} \Vert \partial _x^2w_\varepsilon \Vert _{L^2(\mathbb {R})} \\&\quad + C_1 \Vert \partial _x\rho _\varepsilon \Vert _{L^\infty (\mathbb {R})} \Vert \partial _xw_\varepsilon \Vert _{L^2(\mathbb {R})} \Vert \partial _x^2w_\varepsilon \Vert _{L^2(\mathbb {R})} \le \frac{C_1}{2} \Vert \partial _x^2w_\varepsilon \Vert _{L^2(\mathbb {R})}^2\\&\quad + C \left( \Vert u_\varepsilon \Vert _{L^\infty (\mathbb {R})}^2 + \Vert \partial _x\rho _\varepsilon \Vert _{L^2(\mathbb {R})}^2 + \Vert \partial _x^2\rho _\varepsilon \Vert _{L^2(\mathbb {R})}^2 \right) \Vert \partial _xw_\varepsilon \Vert _{L^2(\mathbb {R})}^2,\\ |I_2|&\le C_1 \Vert \partial _xw_\varepsilon \Vert _{L^2(\mathbb {R})}^2,\\ |I_3|&\le C_1\Vert \partial _x\rho _\varepsilon \Vert _{L^2(\mathbb {R})} \Vert w_\varepsilon \Vert _{L^\infty (\mathbb {R})} \Vert \partial _xw_\varepsilon \Vert _{L^2(\mathbb {R})} \\&\le C_1\Vert \partial _x\rho _\varepsilon \Vert _{L^2(\mathbb {R})} \left( \Vert w_\varepsilon \Vert _{L^\infty (\mathbb {R})}^2 + \Vert \partial _xw_\varepsilon \Vert _{L^2(\mathbb {R})}^2 \right) ,\\ |I_4|&\le 2 C_1 \Vert w_\varepsilon \Vert _{L^\infty (\mathbb {R})} \Vert \partial _xw_\varepsilon \Vert _{L^2(\mathbb {R})}^2,\\ |I_5|&\le C_1 \Vert \partial _x\rho _\varepsilon \Vert _{L^2(\mathbb {R})} \Vert w_\varepsilon \Vert _{L^\infty (\mathbb {R})}^2 \Vert \partial _xw_\varepsilon \Vert _{L^2(\mathbb {R})} \\&\le C_1\Vert \partial _x\rho _\varepsilon \Vert _{L^2(\mathbb {R})} \left( \Vert w_\varepsilon \Vert _{L^\infty (\mathbb {R})}^2 +\Vert w_\varepsilon \Vert _{L^\infty (\mathbb {R})}^2 \Vert \partial _xw_\varepsilon \Vert _{L^2(\mathbb {R})}^2 \right) ,\\ |I_6|&\le C_1\Vert \partial _x\rho _\varepsilon \Vert _{L^2(\mathbb {R})}^2 + C_1 \Vert \partial _xw_\varepsilon \Vert _{L^2(\mathbb {R})}^2. \end{aligned}$$

Moreover, since it follows from (2.7) and $\bar{\rho }\in L^\infty (\mathbb {R})$ that

$$\begin{aligned} \partial _x\rho _\varepsilon \in L^\infty (0,T; L^2(\mathbb {R})) \quad \text{ and }\quad u_\varepsilon \in L^\infty (0,T; L^\infty (\mathbb {R})), \end{aligned}$$

(A.2)

we have

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\Vert \partial _xw_\varepsilon \Vert _{L^2(\mathbb {R})}^2 + C_1 \Vert \partial _x^2w_\varepsilon \Vert _{L^2(\mathbb {R})}^2 \nonumber \\&\quad \le C\left( 1+\Vert \partial _x^2 \rho _\varepsilon \Vert _{L^2(\mathbb {R})}^2 + \Vert w_\varepsilon \Vert _{L^\infty (\mathbb {R})}^2 \right) \Vert \partial _xw_\varepsilon \Vert _{L^2(\mathbb {R})}^2 +F, \end{aligned}$$

(A.3)

where

$$\begin{aligned} F=C\left( 1+\Vert w_\varepsilon \Vert _{L^\infty (\mathbb {R})}^2 \right) . \end{aligned}$$

Note from (A.1) that $F\in L^1((0,T))$.

Step 2 We next estimate $\Vert \partial _x^2 \rho _\varepsilon \Vert _{L^2(\mathbb {R})}$, to control $\Vert \partial _x^2 \rho _\varepsilon \Vert _{L^2(\mathbb {R})}^2$ in (A.3).

Differentiating the continuity equation of (2.6) twice in space, and multiplying the resulting equation by $\partial _x^2\rho _\varepsilon$, we have

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t} \int _\mathbb {R}\frac{|\partial _x^2 \rho _\varepsilon |^2}{2} \mathrm{d}x {=} - \int _\mathbb {R}\partial _x^2(u_\varepsilon \partial _x\rho _\varepsilon ) \partial _x^2\rho _\varepsilon \mathrm{d}x {-} \int _\mathbb {R}\partial _x^2(\rho _\varepsilon \partial _xu_\varepsilon ) \partial _x^2\rho _\varepsilon \mathrm{d}x\\&\quad = -\int _\mathbb {R}u_\varepsilon \partial _x\left( \frac{|\partial _x^2\rho _\varepsilon |^2}{2}\right) \mathrm{d}x -\int _\mathbb {R}\underbrace{\left( \partial _x^2(u_\varepsilon \partial _x\rho _\varepsilon )- u_\varepsilon \partial _x^2 \partial _x\rho _\varepsilon \right) }_{=:J_1} \partial _x^2\rho _\varepsilon \mathrm{d}x \\&\qquad - \int _\mathbb {R}\rho _\varepsilon \partial _x^3 u_\varepsilon \partial _x^2\rho _\varepsilon \mathrm{d}x -\int _\mathbb {R}\underbrace{\left( \partial _x^2(\rho _\varepsilon \partial _xu_\varepsilon ) -\rho _\varepsilon \partial _x^3 u_\varepsilon \right) }_{=:J_2} \partial _x^2\rho _\varepsilon \mathrm{d}x. \end{aligned}$$

Using the commutator estimates (Majda and Bertozzi 2002, Lemma 3.4) and the Sobolev embedding, we have

$$\begin{aligned} \Vert J_1 \Vert _{L^2(\mathbb {R})}&\le C \Vert \partial _x^2 u_\varepsilon \Vert _{L^2(\mathbb {R})} \Vert \partial _x\rho _\varepsilon \Vert _{L^\infty (\mathbb {R})}+ C\Vert \partial _xu_\varepsilon \Vert _{L^\infty (\mathbb {R})} \Vert \partial _x^2 \rho _\varepsilon \Vert _{L^2(\mathbb {R})}\\&\le C \Vert \partial _x^2 u_\varepsilon \Vert _{L^2(\mathbb {R})} \Vert \partial _x\rho _\varepsilon \Vert _{H^1(\mathbb {R})}+ C\Vert \partial _xu_\varepsilon \Vert _{H^1 (\mathbb {R})} \Vert \partial _x^2 \rho _\varepsilon \Vert _{L^2(\mathbb {R})},\\ \Vert J_2 \Vert _{L^2(\mathbb {R})}&\le C \Vert \partial _x^2 \rho _\varepsilon \Vert _{L^2(\mathbb {R})} \Vert \partial _xu_\varepsilon \Vert _{L^\infty (\mathbb {R})}+ C\Vert \partial _x\rho _\varepsilon \Vert _{L^\infty (\mathbb {R})} \Vert \partial _x^2 u_\varepsilon \Vert _{L^2(\mathbb {R})}\\&\le C \Vert \partial _x^2 \rho _\varepsilon \Vert _{L^2(\mathbb {R})} \Vert \partial _xu_\varepsilon \Vert _{H^1(\mathbb {R})}+ C\Vert \partial _x\rho _\varepsilon \Vert _{H^1 (\mathbb {R})} \Vert \partial _x^2 u_\varepsilon \Vert _{L^2(\mathbb {R})}. \end{aligned}$$

Therefore, we have

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t} \int _\mathbb {R}\frac{|\partial _x^2 \rho _\varepsilon |^2}{2} \mathrm{d}x {\le } \frac{1}{2} \Vert \partial _xu_\varepsilon \Vert _{L^\infty (\mathbb {R})} \Vert \partial _x^2 \rho _\varepsilon \Vert _{L^2(\mathbb {R})}^2 {+} \Vert \rho _\varepsilon \Vert _{L^\infty (\mathbb {R})} \Vert \partial _x^3 u_\varepsilon \Vert _{L^2(\mathbb {R})} \Vert \partial _x^2 \rho _\varepsilon \Vert _{L^2(\mathbb {R})}\\&\quad +\,C\left( \Vert \partial _x^2 u_\varepsilon \Vert _{L^2(\mathbb {R})} \Vert \partial _x\rho _\varepsilon \Vert _{L^2(\mathbb {R})} +\Vert \partial _xu_\varepsilon \Vert _{H^1(\mathbb {R})} \Vert \partial _x^2 \rho _\varepsilon \Vert _{L^2(\mathbb {R})} \right) \Vert \partial _x^2 \rho _\varepsilon \Vert _{L^2(\mathbb {R})}. \end{aligned}$$

Moreover, using (2.8), (A.2) and the Sobolev embedding, we have

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} \Vert \partial _x^2 \rho _\varepsilon \Vert _{L^2(\mathbb {R})}^2&\le C\left( \Vert \partial _x u_\varepsilon \Vert _{H^1(\mathbb {R})} + \Vert \partial _x^2 u_\varepsilon \Vert _{L^2(\mathbb {R})}^2 \right) \Vert \partial _x^2 \rho _\varepsilon \Vert _{L^2(\mathbb {R})}^2 \nonumber \\&\quad +\,C \Vert \partial _x^3 u_\varepsilon \Vert _{L^2(\mathbb {R})} \Vert \partial _x^2 \rho _\varepsilon \Vert _{L^2(\mathbb {R})} +C. \end{aligned}$$

(A.4)

To estimate $\Vert \partial _x^3 u_\varepsilon \Vert _{L^2(\mathbb {R})}$ in (A.4), we use the definition (2.9) of $w_\varepsilon$ as follows:

$$\begin{aligned} \partial _xu_\varepsilon = g(\rho _\varepsilon ) w_\varepsilon + h(\rho _\varepsilon ),\quad \text{ where } ~ g(\rho _\varepsilon ):=\frac{1}{\mu _\varepsilon (\rho _\varepsilon )},~h(\rho _\varepsilon ):=\frac{p(\rho _\varepsilon )}{\mu _\varepsilon (\rho _\varepsilon )}. \end{aligned}$$

(A.5)

Since

$$\begin{aligned} \partial _x^3 u_\varepsilon&=g''(\rho _\varepsilon ) |\partial _x\rho _\varepsilon |^2 w_\varepsilon + g'(\rho _\varepsilon ) \partial _x^2\rho _\varepsilon w_\varepsilon +2g'(\rho _\varepsilon ) \partial _x\rho _\varepsilon \partial _xw_\varepsilon + g(\rho _\varepsilon ) \partial _x^2 w_\varepsilon \\&\quad +\,h''(\rho _\varepsilon ) |\partial _x\rho _\varepsilon |^2 + h'(\rho _\varepsilon ) \partial _x^2\rho _\varepsilon , \end{aligned}$$

we use (2.8) to have

$$\begin{aligned} \Vert \partial _x^3 u_\varepsilon \Vert _{L^2(\mathbb {R})}&\le C\Big ( \big (\Vert w_\varepsilon \Vert _{L^\infty (\mathbb {R})} +1\big ) \Vert \partial _x\rho _\varepsilon \Vert _{L^\infty (\mathbb {R})} \Vert \partial _x\rho _\varepsilon \Vert _{L^2(\mathbb {R})} \nonumber \\&\quad + \Vert w_\varepsilon \Vert _{L^\infty (\mathbb {R})} \Vert \partial _x^2\rho _\varepsilon \Vert _{L^2(\mathbb {R})} \nonumber \\&\quad +\,\Vert \partial _x\rho _\varepsilon \Vert _{L^\infty (\mathbb {R})} \Vert \partial _xw_\varepsilon \Vert _{L^2(\mathbb {R})} + \Vert \partial _x^2w_\varepsilon \Vert _{L^2(\mathbb {R})} + \Vert \partial _x^2\rho _\varepsilon \Vert _{L^2(\mathbb {R})} \Big ). \end{aligned}$$

(A.6)

Combining this with (A.4), and using (A.2) and the Sobolev embedding, we have

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} \Vert \partial _x^2 \rho _\varepsilon \Vert _{L^2(\mathbb {R})}^2 \le \frac{C_1}{2} \Vert \partial _x^2w_\varepsilon \Vert _{L^2(\mathbb {R})}^2 +G_1 \Vert \partial _x^2 \rho _\varepsilon \Vert _{L^2(\mathbb {R})}^2 + G_2, \end{aligned}$$

(A.7)

where

$$\begin{aligned} G_1&:= C \left( \Vert \partial _x u_\varepsilon \Vert _{H^1(\mathbb {R})} + \Vert \partial _x^2 u_\varepsilon \Vert _{L^2(\mathbb {R})}^2 +\Vert w_\varepsilon \Vert _{L^\infty (\mathbb {R})} + \Vert \partial _xw_\varepsilon \Vert _{L^2(\mathbb {R})} +1 \right) ,\\ G_2&:= C \left( \Vert w_\varepsilon \Vert _{L^\infty (\mathbb {R})}^2 + \Vert \partial _xw_\varepsilon \Vert _{L^2(\mathbb {R})}^2 +1 \right) . \end{aligned}$$

Note that $G_1, G_2 \in L^1((0,T))$ by (2.7) and (A.1).

Step 3 Adding (A.3)–(A.7), we have

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left( \Vert \partial _xw_\varepsilon \Vert _{L^2(\mathbb {R})}^2+\Vert \partial _x^2 \rho _\varepsilon \Vert _{L^2(\mathbb {R})}^2 \right) + \frac{C_1}{2} \Vert \partial _x^2w_\varepsilon \Vert _{L^2(\mathbb {R})}^2 \\&\quad \le H \left( \Vert \partial _xw_\varepsilon \Vert _{L^2(\mathbb {R})}^2+\Vert \partial _x^2 \rho _\varepsilon \Vert _{L^2(\mathbb {R})}^2 \right) + F+G_2, \end{aligned}$$

where

$$\begin{aligned} H:=C\left( 1+\Vert \partial _xw_\varepsilon \Vert _{L^2(\mathbb {R})}^2+ \Vert w_\varepsilon \Vert _{L^\infty (\mathbb {R})}^2 + \Vert \partial _x u_\varepsilon \Vert _{H^1(\mathbb {R})} + \Vert \partial _x^2 u_\varepsilon \Vert _{L^2(\mathbb {R})}^2 \right) . \end{aligned}$$

Since $H, F, G_2\in L^1((0,T))$, and it follows from (2.9) and (2.11) that

$$\begin{aligned}&\Vert \partial _xw_\varepsilon (0)\Vert _{L^2(\mathbb {R})} \\&\quad \le C(\underline{\kappa }_0, \overline{\kappa }_0)\left( \Vert \partial _x\rho _0\Vert _{L^2(\mathbb {R})} + \Vert \partial _x\rho _0\Vert _{L^2(\mathbb {R})} \Vert \partial _xu_0\Vert _{L^2(\mathbb {R})}+\Vert \partial _x^2 u_0\Vert _{L^2(\mathbb {R})} \right) , \end{aligned}$$

Grönwall lemma implies that

$$\begin{aligned} \Vert \partial _x^2 \rho _\varepsilon \Vert _{L^\infty (0,T;L^2(\mathbb {R}))} + \Vert \partial _xw_\varepsilon \Vert _{L^\infty (0,T;L^2(\mathbb {R}))}+\Vert \partial _x^2 w_\varepsilon \Vert _{L^2(0,T;L^2(\mathbb {R}))} \le C, \end{aligned}$$

(A.8)

where the constant $C>0$ depends on T and the bounds of (2.7), (2.8) and (2.11).

This now together with (A.1), (A.2) and (A.6) imply the bound for $\partial _x^3 u_\varepsilon$:

$$\begin{aligned} \Vert \partial _x^3 u_\varepsilon \Vert _{L^2(0,T;L^2(\mathbb {R}))} \le C. \end{aligned}$$

Moreover, differentiating the both sides of (A.5) in x, and using (2.8), we have

$$\begin{aligned} \Vert \partial _x^2 u_\varepsilon \Vert _{L^2(\mathbb {R})} \le C\Big ( \Vert \partial _x\rho _\varepsilon \Vert _{L^2(\mathbb {R})} \Vert w_\varepsilon \Vert _{L^\infty (\mathbb {R})}+ \Vert \partial _xw_\varepsilon \Vert _{L^2(\mathbb {R})} + \Vert \partial _x\rho _\varepsilon \Vert _{L^2(\mathbb {R})} \Big ). \end{aligned}$$

Therefore, we use (2.7), (2.8) and (A.8) to have

$$\begin{aligned} \Vert \partial _x^2 u_\varepsilon \Vert _{L^\infty (0,T;L^2(\mathbb {R}))} \le C. \end{aligned}$$

Indeed, since it follows from (2.7) and (2.8) that

$$\begin{aligned} w_\varepsilon = -p(\rho _\varepsilon ) +\mu _\varepsilon (\rho _\varepsilon )\partial _xu_\varepsilon \in L^\infty ((0,T)\times \mathbb {R})+ L^\infty (0,T;L^2(\mathbb {R})), \end{aligned}$$

we use (A.8) to have

$$\begin{aligned} |w_\varepsilon (x)|&\le \frac{1}{2}\int _{x-1}^{x+1} (|p(\rho _\varepsilon )| +|\mu _\varepsilon (\rho _\varepsilon )\partial _xu_\varepsilon |) \mathrm{d}y + \frac{1}{2}\int _{x-1}^{x+1}\int _y^x |\partial _z w_\varepsilon | \mathrm{d}z \mathrm{d}y\\&\le \Vert p(\rho _\varepsilon ) \Vert _{L^\infty ((0,T)\times \mathbb {R})} + \frac{1}{\sqrt{2}} \Vert \mu _\varepsilon (\rho _\varepsilon )\partial _xu_\varepsilon \Vert _{L^\infty (0,T;L^2(\mathbb {R}))} \\&\quad +\,\sqrt{2} \Vert \partial _xw_\varepsilon \Vert _{L^\infty (0,T;L^2(\mathbb {R}))}, \end{aligned}$$

which gives $\Vert w_\varepsilon \Vert _{L^\infty ((0,T)\times \mathbb {R})}\le C$.

Hence, we complete the proof.