Existence of global weak solutions for 3D degenerate compressible Navier–Stokes equations

Abstract

In this paper, we prove the existence of global weak solutions for 3D compressible Navier–Stokes equations with degenerate viscosity. The method is based on the Bresch and Desjardins (Commun Math Phys 238:211–223 2003) entropy conservation. The main contribution of this paper is to derive the Mellet and Vasseur (Commun Partial Differ Equ 32:431–452, 2007) type inequality for weak solutions, even if it is not verified by the first level of approximation. This provides existence of global solutions in time, for the compressible barotropic Navier–Stokes equations. The result holds for any $\gamma >1$ in two dimensional space, and for $1<\gamma <3$ in three dimensional space, in both case with large initial data possibly vanishing on the vacuum. This solves an open problem proposed by Lions (Mathematical topics in fluid mechanics. Vol. 2. Compressible models, 1998).

Appendix1: Proof of the Lemma 2.2

Proof

We prove each statement one by one as follows:

• (a) Thanks to (2.4), we have $\varphi '_n(\mathbf{u})=2\tilde{\varphi }'_n(|\mathbf{u}|^2)\mathbf{u}$, and

  $$\begin{aligned} \varphi _n''(\mathbf{u})=2\left( 2\tilde{\varphi }_n''(|\mathbf{u}|^2)\mathbf{u}\otimes \mathbf{u}+\mathbf {I} \tilde{\varphi }_n'(|\mathbf{u}|^2)\right) , \end{aligned}$$

  where $\mathbf {I}$ is $3\times 3$ identity matrix.

• (b) The statement of (b) follows directly from (2.5).

• (c) Integrating (2.5) with initial data $\tilde{\varphi }'_n(0)=0,$ one obtains

  $$\begin{aligned} \tilde{\varphi }'_n(y)=\left\{ \begin{array}{ll} 1+\ln (1+y) &{}\quad \text { if }\,0\le y< n, \\ 1+2\ln (1+n)-\ln (1+y), &{}\quad \text { if }\, n\le y\le C_n \\ 0&{}\quad \text { if } y \ge C_n, \end{array}\right. \end{aligned}$$

  (7.1)

  Since

  $$\begin{aligned} 1+2\ln (1+n)-\ln (1+C_n)= 0, \end{aligned}$$

  for any $y\ge 0$, (7.1) implies

  $$\begin{aligned} \tilde{\varphi }'_n(y)\ge 0. \end{aligned}$$

  For any $n\le y\le C_n,$ we have

  $$\begin{aligned} 1+2\ln (1+n)-\ln (1+y)\le & {} 1+2\ln (1+y)-\ln (1+y)\\= & {} 1+\ln (1+y). \end{aligned}$$

  In one word, for any $y\ge 0$, we have

  $$\begin{aligned} 0\le \tilde{\varphi }'_n(y)\le 1+\ln (1+y). \end{aligned}$$

• (d) By (a)–(c), it follows

  $$\begin{aligned} |\varphi ^{''}_n(\mathbf{u})|\le & {} 4|\tilde{\varphi }''_n||\mathbf{u}|^2+2|\tilde{\varphi }'_n|\le 4\frac{|\mathbf{u}|^2}{1+|\mathbf{u}|^2}+2(1+\ln (1+n))\\\le & {} 6+2\ln (1+n). \end{aligned}$$

• (e) Integrating (7.1) with initial data $\tilde{\varphi }_n(0)=0$, it gives (2.9). Moreover, thanks to (c), $\tilde{\varphi }_n(y)$ is an increasing function with respect to y for any fixed n. We have that $\tilde{\varphi }_n(y)$ is a nondecreasing function with respect to n for any fixed y.

$\square$