Global smooth solutions for 1D barotropic Navier-Stokes equations with a large class of degenerate viscosities

TL;DR

This paper proves the global existence and uniqueness of smooth solutions for the 1D barotropic Navier-Stokes equations with a broad class of degenerate viscosities, including models like viscous shallow water equations, extending previous periodic domain results.

Contribution

It establishes the global well-posedness of smooth solutions with possibly different far-fields for all positive alpha, broadening the class of degenerate viscosities covered.

Findings

Global existence and uniqueness of smooth solutions.

Solutions can connect two different far-field states.

Results apply to a large class of degenerate viscosities including shallow water models.

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-Drivas-Nguyen-Pasqualotto \cite[Theorem 1.5]{CDNP} (on the case of periodic domain) to the case where smooth solutions connect possibly two different limits at the infinity on the whole space.

Equations

$$\displaystyle\begin{aligned} \left\{\begin{array}[]{ll}\rho_{t}+(\rho u)_{x}=0,\\ (\rho u)_{t}+(\rho u^{2})_{x}+p(\rho)_{x}=(\mu(\rho)u_{x})_{x},\end{array}\right.\end{aligned}$$

$$p(\rho )={\rho }^{\gamma },\gamma >1,$$

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

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

$$\displaystyle\begin{aligned} &\rho_{0}-\bar{\rho}\in H^{k}(\mathbb{R}),\qquad u_{0}-\bar{u}\in H^{k}(\mathbb{R}),\qquad\mbox{for some integer }k\geq 4,\\ &0<\underline{\kappa}_{0}\leq\rho_{0}(x)\leq\overline{\kappa}_{0},\quad\forall x\in\mathbb{R},\qquad\mbox{for some constants $\underline{\kappa}_{0},\overline{\kappa}_{0}$},\end{aligned}$$

$${\partial }_x{u}_0(x)\le {\rho }_0(x{)}^{\gamma -\alpha },\forall x\in \mathbb{R},$$

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

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

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

$$\displaystyle\begin{aligned} \nu(y)\geq\left\{\begin{array}[]{ll}Cy^{q}&\forall y\leq 1\\ C&\forall y\geq 1,\end{array}\right.\end{aligned}$$

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

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

$${\mu }_{\epsilon }(\rho ):=\max \left(\mu (\rho ),\epsilon {\rho }^{{\alpha }_{\ast }}\right),\forall \rho \ge 0,\text{mbox}where{\alpha }_{\ast }:=\frac{1}{2}\min \left(\alpha ,\frac{1}{2}\right).$$

$$\mu_{\varepsilon}(\rho)\geq\left\{\begin{array}[]{ll}\varepsilon\rho^{1/4}&\forall\rho\leq 1\\ \varepsilon&\forall\rho\geq 1,\end{array}\right.$$

$${\mu }_{\epsilon }(\rho )\le 1+{\rho }^{\gamma }\forall \rho \ge 0,$$

$$\displaystyle\begin{aligned} \left\{\begin{array}[]{ll}\partial_{t}\rho_{\varepsilon}+\partial_{x}(\rho_{\varepsilon}u_{\varepsilon})=0\\ \partial_{t}(\rho_{\varepsilon}u_{\varepsilon})+\partial_{x}(\rho_{\varepsilon}u_{\varepsilon}^{2})+\partial_{x}p(\rho_{\varepsilon})=\partial_{x}(\mu_{\varepsilon}(\rho_{\varepsilon})\partial_{x}u_{\varepsilon})\\ (\rho_{\varepsilon},u_{\varepsilon})|_{t=0}=(\rho_{0},u_{0}),\end{array}\right.\end{aligned}$$

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

$${\underline{\kappa }}_{\epsilon }(T)\le {\rho }_{\epsilon }(t,x)\le {\overline{\kappa }}_{\epsilon }(T),\forall (t,x)\in (0,T)\times \mathbb{R}.$$

$$w_{\epsilon }:=-p({\rho }_{\epsilon })+{\mu }_{\epsilon }({\rho }_{\epsilon }){\partial }_x{u}_{\epsilon }.$$

$${\rho }_{\epsilon }({\partial }_t{u}_{\epsilon }+u_{\epsilon }{\partial }_x{u}_{\epsilon })={\partial }_x{w}_{\epsilon }.$$

$$\begin{array}{rl}{\partial }_t{w}_{\epsilon }& =\frac{{\mu }_{\epsilon }({\rho }_{\epsilon })}{{\rho }_{\epsilon }}{\partial }_x^2{w}_{\epsilon }-\left(u_{\epsilon }+{\mu }_{\epsilon }({\rho }_{\epsilon })\frac{{\partial }_x{\rho }_{\epsilon }}{{\rho }_{\epsilon }^2}\right){\partial }_x{w}_{\epsilon }+\left({\rho }_{\epsilon }\frac{p^{\prime }({\rho }_{\epsilon })}{{\mu }_{\epsilon }({\rho }_{\epsilon })}-2p({\rho }_{\epsilon })\frac{{\rho }_{\epsilon }{\mu }_{\epsilon }^{\prime }({\rho }_{\epsilon })+{\mu }_{\epsilon }({\rho }_{\epsilon })}{{\mu }_{\epsilon }({\rho }_{\epsilon }{)}^2}\right)w_{\epsilon }\\ & -\frac{{\rho }_{\epsilon }{\mu }_{\epsilon }^{\prime }({\rho }_{\epsilon })+{\mu }_{\epsilon }({\rho }_{\epsilon })}{{\mu }_{\epsilon }({\rho }_{\epsilon }{)}^2}{w}_{\epsilon }^2+\left({\rho }_{\epsilon }\frac{p^{\prime }({\rho }_{\epsilon })}{{\mu }_{\epsilon }({\rho }_{\epsilon })}-p({\rho }_{\epsilon })\frac{{\rho }_{\epsilon }{\mu }_{\epsilon }^{\prime }({\rho }_{\epsilon })+{\mu }_{\epsilon }({\rho }_{\epsilon })}{{\mu }_{\epsilon }({\rho }_{\epsilon }{)}^2}\right)p({\rho }_{\epsilon }).\end{array}$$

$$\begin{array}{rl}& {\rho }_0-\bar{\rho }\in H^k(\mathbb{R}),u_0-\bar{u}\in H^k(\mathbb{R}),\text{mbox}forsomeintegerk\ge 2,\\ & 0<{\underline{\kappa }}_0\le {\rho }_0(x)\le {\overline{\kappa }}_0,\forall x\in \mathbb{R},\end{array}$$

$$\begin{array}{rl}\Vert {\partial }_x^k{\rho }_{\epsilon }{\Vert }_{L^{\infty }(0,T;L^2(\mathbb{R}))}& +\Vert {\partial }_x^{k-1}{w}_{\epsilon }{\Vert }_{L^{\infty }(0,T;L^2(\mathbb{R}))}+\Vert {\partial }_x^k{w}_{\epsilon }{\Vert }_{L^2(0,T;L^2(\mathbb{R}))}\\ & +\Vert {\partial }_x^k{u}_{\epsilon }{\Vert }_{L^{\infty }(0,T;L^2(\mathbb{R}))}+\Vert {\partial }_x^{k+1}{u}_{\epsilon }{\Vert }_{L^2(0,T;L^2(\mathbb{R}))}\le C.\end{array}$$

$$\begin{array}{r}{w}_{\epsilon }(t,x)\le C_{\gamma }{\epsilon }^{\theta },\forall \epsilon \le {\epsilon }_{\gamma },\forall t\le T,\forall x\in \mathbb{R},\end{array}$$

$$\displaystyle\begin{aligned} \theta:=\frac{\gamma}{\alpha-\alpha_{*}},\qquad\mbox{where $\alpha_{*}$ is the constant as in \eqref{new-v}}.\end{aligned}$$

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

$$w_{\epsilon }(0,x)=-p({\rho }_0)+\max \left(\mu ({\rho }_0),\epsilon {\rho }_0^{{\alpha }_{\ast }}\right){\partial }_x{u}_0\le -{\rho }_0^{\gamma }+\max \left({\rho }_0^{\alpha },\epsilon {\rho }_0^{{\alpha }_{\ast }}\right){\rho }_0^{\gamma -\alpha }.$$

$$\begin{array}{rl}{w}_{\epsilon }(0,x)& \le \left(-{\rho }_0^{\gamma }+{\rho }_0^{\alpha }{\rho }_0^{\gamma -\alpha }\right)1_{\{{\rho }_0^{\alpha }>\epsilon {\rho }_0^{{\alpha }_{\ast }}\}}+\left(-{\rho }_0^{\gamma }+\epsilon {\rho }_0^{{\alpha }_{\ast }}{\rho }_0^{\gamma -\alpha }\right)1_{\{{\rho }_0^{\alpha }\le \epsilon {\rho }_0^{{\alpha }_{\ast }}\}}\\ & \le \epsilon {\rho }_0^{\gamma -(\alpha -{\alpha }_{\ast })}{1}_{\{{\rho }_0^{\alpha }\le \epsilon {\rho }_0^{{\alpha }_{\ast }}\}}\le {\epsilon }^{\frac{\gamma }{\alpha -{\alpha }_{\ast }}},\end{array}$$

$$w_{\epsilon }(0,x)\le {\epsilon }^{\theta },\forall x\in \mathbb{R}.$$

$$\underset{x\in \mathbb{R}}{\sup }{w}_{\epsilon }(t,x)\le {\epsilon }^{\theta }\forall t\in [0,t_1],$$

Full text

Acknowledgment. The first author was partially supported by the NRF-2019R1C1C1009355. The second author was partially supported by the NSF grant: DMS 1614918.

1. 1 Introduction

2. 1.1 Main results

3. 2 Proof of Theorem 1.1

4. 2.1 Idea of Proof

5. 2.2 Approximate viscosity

6. 2.3 Higher Sobolev regularity

7. 2.4 Uniform lower bound for the density

8. 2.5 Uniform bounds for the solutions $(\rho_{\varepsilon},u_{\varepsilon})$

9. 2.6 Conclusion

10. A Proof of Lemma 2.1

1. Introduction

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

[TABLE]

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

[TABLE]

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

[TABLE]

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 [4]). 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 [13] 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 [17, 18, 20] and then by Serre [16] and Hoff [8]. First result for vanishing initial density was obtained by Shelukhin [19]. Hoff [9] 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 was first studied by Solonnikov [21] for smooth initial data and for small time. However, the regularity may blow-up as the solution gets close to vacuum. Hoff and Smoller [10] show that any weak solution of the one-dimensional Navier-Stokes equations do 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-Vasseur [15] 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 [1] for the multi-dimensional Korteweg system of equations (for the case of $\alpha=1$ and with an additional capillary term) and later generalized in [3]. In the one-dimensional case, a similar inequality was introduced earlier by Vaigant [22] for flows with constant viscosity.

The result of Mellet-Vasseur [15] was extended by Haspot [7] to the case of $\alpha\in(1/2,1]$. Recently, Constantin-Drivas-Nguyen-Pasqualotto [5, Theorem 1.6] extended it to the case of $\alpha\geq 0$ 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 [5, 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 [11, 12] of the authors on the contraction property for any large perturbations of viscous shocks of the one-dimensional barotropic Navier-Stokes system with degenerate viscosity.

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

[TABLE]

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

[TABLE]

and

[TABLE]

*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 $T>0$,*

[TABLE]

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

[TABLE]

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 [11, 12] for viscous shocks of the barotropic Navier-Stokes system in the mass Lagrangian coordinates, with any large initial data satisfying (LABEL:ass-ini) 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-Perthame [6] 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 $k\geq 4$ in (LABEL:ass-ini) are the same conditions as in [5, Theorem 1.5], which is used to control the active potential (2.9) defined by the density and the velocity (see Lemma 2.2).

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 [15], 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 [15] 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 [5] 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 (LABEL:ass-ini).

2.2. Approximate viscosity

As mentioned above, we first recall the existence result in [15] as follows:

Proposition 2.1**.**

[15]** Let $\rho_{0}$ and $u_{0}$ be the initial data such that

[TABLE]

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

[TABLE]

and

[TABLE]

*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*

[TABLE]

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 $0<\varepsilon<1$,

[TABLE]

Since

[TABLE]

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

[TABLE]

$\mu_{\varepsilon}$ satisfies the assumptions (2.2) and (2.3). Therefore, for the initial datum $(\rho_{0},u_{0})$ satisfying (LABEL:ass-ini), 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.,

[TABLE]

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

[TABLE]

and

[TABLE]

2.3. Higher Sobolev regularity

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

[TABLE]

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

[TABLE]

Then it follows from [5, Proposition 3.1] that $w_{\varepsilon}$ satisfies a forced quadratic heat equation with linear drift:

[TABLE]

Note that the new viscosity coefficient $\mu_{\varepsilon}(\rho_{\varepsilon})/\rho_{\varepsilon}$ of the parabolic equation (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 $\eqref{NS-eps}_{1}$, 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

[TABLE]

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

[TABLE]

This follows straightforwardly from [5, Lemma 4.2 and 4.3] when $\|w_{\varepsilon}\|_{L^{\infty}(0,T;L^{2}(\mathbb{R}))}$ 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 [5], 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

[TABLE]

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

[TABLE]

Proof.

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

[TABLE]

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

[TABLE]

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

[TABLE]

we have

[TABLE]

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 $t_{1}\geq 0$ such that

[TABLE]

and

[TABLE]

Let

[TABLE]

Then,

[TABLE]

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

[TABLE]

we can define the function

[TABLE]

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

[TABLE]

Then $w^{\prime}_{M}(t)=(\partial_{t}w_{\varepsilon})(t,x_{t})$ for a.e. $t\in(t_{1},t_{2})$, since

[TABLE]

Using this together with $\partial_{x}^{2}w_{\varepsilon}(t,x_{t})\leq 0$, $\partial_{x}w_{\varepsilon}(t,x_{t})=0$ and $\rho_{\varepsilon}\mu_{\varepsilon}^{\prime}(\rho_{\varepsilon})\geq 0$, we have from (2.10) that

[TABLE]

where (putting $\rho_{M}(t):=\rho_{\varepsilon}(t,x_{t})$)

[TABLE]

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

[TABLE]

Moreover, using $\mu_{\varepsilon}(\rho_{M})\geq\varepsilon\rho_{M}^{\alpha_{*}}$ and $\mu_{\varepsilon}(\rho_{M})\geq\rho_{M}^{\alpha}$ by the definition, we have

[TABLE]

Likewise, we have

[TABLE]

The above estimates and (2.13) imply that for any $t\in[t_{1},t_{2}]$ and $\varepsilon\in(0,1)$,

[TABLE]

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

[TABLE]

the following holds:

[TABLE]

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

[TABLE]

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

[TABLE]

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

If $t_{2}<T$, then the definition of $t_{2}$ implies

[TABLE]

Hence we complete the proof. ∎

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

[TABLE]

Proof.

Let

[TABLE]

We first choose a constant $\delta_{1}>0$ such that

[TABLE]

where $\underline{\kappa}_{0}$ is the constant as in (LABEL:ass-ini), and $\varepsilon_{\gamma},C_{\gamma}$ are the constants as in Lemma 2.2.

Then, since

[TABLE]

we have $2\delta_{1}^{q(\gamma)/\gamma}<\underline{\kappa}_{0}$ for any $\gamma\geq\alpha$.

Therefore, it follows from the initial condition of (LABEL:ass-ini) that

[TABLE]

For any fixed $\varepsilon\leq\delta_{1}$, 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 $t_{1}\geq 0$ such that

[TABLE]

[TABLE]

Then,

[TABLE]

where

[TABLE]

Thus, using $2\delta_{1}^{q(\gamma)/\gamma}<\underline{\kappa}_{0}\leq\min(\rho_{-},\rho_{+})$ together with the fact that for each $t\leq T$,

[TABLE]

we define the function

[TABLE]

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 $\rho^{\prime}_{m}(t)=(\partial_{t}\rho_{\varepsilon})(t,y_{t})$ 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

[TABLE]

Then, using (2.9), Lemma 2.2 with $\varepsilon\leq\delta_{1}\leq\varepsilon_{\gamma}$, and $\mu_{\varepsilon}(\rho_{m})\geq\rho_{m}^{\alpha}$, we have

[TABLE]

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

[TABLE]

which yields

[TABLE]

Thus, using (2.16), we have

[TABLE]

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

[TABLE]

we have

[TABLE]

Therefore, this together with (2.16) and the definition of $t_{2}$ implies

[TABLE]

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

[TABLE]

Then, since

[TABLE]

we have

[TABLE]

which together with (2.16) yields

[TABLE]

Since $q(\gamma)/\gamma=1/\alpha$ and $\theta=\alpha/(\alpha-\alpha_{*})$ when $\gamma=\alpha$, if needed, taking $\delta_{1}$ again such that

[TABLE]

we have

[TABLE]

Therefore, this together with (2.16) and the definition of $t_{2}$ implies

[TABLE]

Hence we complete the proof. ∎

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

[TABLE]

where $\delta_{1}$ is the constant as in Proposition 2.2.

For the proof of Proposition 2.3, we refer to the proof of [15, Proposition 4.5], in which the uniform estimates (2.19) and (LABEL:2est-bd) 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

[TABLE]

The others are the estimates [15, Lemmas 3.1 and 3.2] on the relative entropy related to the Bresch-Desjardins entropy (see [1, 2, 3]) as follows:

[TABLE]

where $\varphi^{\prime}(\rho_{\varepsilon}):=\mu_{\varepsilon}(\rho_{\varepsilon})/\rho_{\varepsilon}^{2}$, and the above constant $K$ is independent of $\varepsilon$ thanks to (2.5). Indeed, it follows from [15, 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)-(LABEL:2est-bd) 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

[TABLE]

For the proof of proposition 2.4, we first refer to the proof of [15, 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 $\varepsilon\leq\delta_{1}$, the system (2.6) has the unique smooth solution $(\rho_{\varepsilon},u_{\varepsilon})$ such that Propositions 2.2, 2.3 and 2.4 hold.

We now take $\delta_{T}$ as

[TABLE]

where the constants $\underline{\kappa}(T)$ and $\delta_{1}$ are as in Proposition 2.2.

Then, since Proposition 2.2 implies that for all $\varepsilon<\delta_{T}$,

[TABLE]

it follows from the definition (1.3) that

[TABLE]

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, 2.3 and 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 [5, Lemma 4.3]. Therefore, we here present the proof only when $k=2$, based on the proof of [5, 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

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Step 1) Differentiating the equation (2.10) in space, multiplying the resulting equation by $\partial_{x}w_{\varepsilon}$ and integrating by parts, we have

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where

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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

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and

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Thus, the above terms $I_{j}$ can be controlled as follows:

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

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we have

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where

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Note from (LABEL:reg-w) that $F\in L^{1}((0,T))$.

Step 2) We next estimate $\|\partial_{x}^{2}\rho_{\varepsilon}\|_{L^{2}(\mathbb{R})}$, to control $\|\partial_{x}^{2}\rho_{\varepsilon}\|_{L^{2}(\mathbb{R})}^{2}$ 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

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Using the commutator estimates [14, Lemma 3.4] and the Sobolev embedding, we have

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Therefore, we have

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

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

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Since

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we use (2.8) to have

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

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where

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Note that $G_{1},G_{2}\in L^{1}((0,T))$ by (2.7) and (LABEL:reg-w).

Step 3) Adding (A.3) to (A.7), we have

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where

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Since $H,F,G_{2}\in L^{1}((0,T))$, and it follows from (2.9) and (LABEL:temp-ini) that

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Gr$\ddot{\mbox{o}}$nwall lemma implies that

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where the constant $C>0$ depends on $T$ and the bounds of (2.7), (2.8) and (LABEL:temp-ini).

This now together with (LABEL:reg-w), (A.2) and (A.6) imply the bound for $\partial_{x}^{3}u_{\varepsilon}$:

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

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

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

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we use (A.8) to have

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

Hence we complete the proof.