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Global stability for compressible isentropic Navier-Stokes equations in 3D bounded domains with Navier-slip boundary conditions

Yang Liu, Guochun Wu, Xin Zhong

Abstract

We investigate the global stability of large solutions to the compressible isentropic Navier-Stokes equations in a three-dimensional (3D) bounded domain with Navier-slip boundary conditions. It is shown that the strong solutions converge to an equilibrium state exponentially in the $L^2$-norm provided the density is essentially uniform-in-time bounded from above. Moreover, we obtain that the density converges to its equilibrium state exponentially in the $L^\infty$-norm if additionally the initial density is bounded away from zero. Furthermore, we derive that the vacuum states will not vanish for any time provided vacuum appears (even at a point) initially. This is the first result concerning the global stability for large strong solutions of compressible Navier-Stokes equations with vacuum in 3D general bounded domains.

Global stability for compressible isentropic Navier-Stokes equations in 3D bounded domains with Navier-slip boundary conditions

Abstract

We investigate the global stability of large solutions to the compressible isentropic Navier-Stokes equations in a three-dimensional (3D) bounded domain with Navier-slip boundary conditions. It is shown that the strong solutions converge to an equilibrium state exponentially in the -norm provided the density is essentially uniform-in-time bounded from above. Moreover, we obtain that the density converges to its equilibrium state exponentially in the -norm if additionally the initial density is bounded away from zero. Furthermore, we derive that the vacuum states will not vanish for any time provided vacuum appears (even at a point) initially. This is the first result concerning the global stability for large strong solutions of compressible Navier-Stokes equations with vacuum in 3D general bounded domains.

Paper Structure

This paper contains 3 sections, 14 theorems, 130 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. The proofs of Theorems \ref{['thm1']} and \ref{['thm2']}

Key Result

Theorem 1.1

Assume that the initial data $(\rho_0\ge 0, u_0)$ satisfies Let $(\rho, u)$ be a global strong solution to the problem a1--a4 verifying that for some positive constant $\hat{\rho}$. Then there exist some positive constants $C_1$ and $\eta_1$, which are dependent on $\hat{\rho}$ and $K$, but independent of $t$, such that If additionally $\inf\limits_{x\in\Omega}\rho_0(x)\ge \rho_*>0$, then there

Theorems & Definitions (29)

  • Definition 1.1: Strong solutions
  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.2
  • Remark 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • ...and 19 more