Free boundary problem for compressible  flows with density--dependent viscosity coefficients

Ping Chen; Daoyuan Fang; Ting Zhang

doi:10.3934/cpaa.2011.10.459

Article Contents

2011, Volume 10, Issue 2: 459-478. Doi: 10.3934/cpaa.2011.10.459

This issue Previous Article The Boltzmann equation near Maxwellian in the whole space Next Article A system of the Hamilton--Jacobi and the continuity equations in the vanishing viscosity limit

Free boundary problem for compressible flows with density--dependent viscosity coefficients

1.
Department of Mathematics, Zhejiang University, Hangzhou 310027

Received: April 30, 2010

Revised: August 31, 2010

Published: February 2011

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Abstract

In this paper, we consider the free boundary problem of the spherically symmetric compressible isentropic Navier--Stokes equations in $R^n (n \geq 1)$, with density--dependent viscosity coefficients. Precisely, the viscosity coefficients $\mu$ and $\lambda$ are assumed to be proportional to $\rho^\theta$, $0 < \theta < 1$, where $\rho$ is the density. We obtain the global existence, uniqueness and continuous dependence on initial data of a weak solution, with a Lebesgue initial velocity $u_0\in L^{4 m}$, $4m>n$ and $\theta<\frac{4m-2}{4m+n}$. We weaken the regularity requirement of the initial velocity, and improve some known results of the one-dimensional system.

Keywords:

Compressible Navier-Stokes equations,
density-dependent viscosity coefficients.

Mathematics Subject Classification: Primary: 76D05, 35R35; Secondary: 35Q35, 76N10.

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