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

Pages: 459 - 478, Volume 10, Issue 2, March 2011

doi:10.3934/cpaa.2011.10.459       Abstract        References        Full Text (443.8K)       Related Articles

Ping Chen - Department of Mathematics, Zhejiang University, Hangzhou 310027, China (email)
Daoyuan Fang - Department of Mathematics, Zhejiang University, Hangzhou 310027, China (email)
Ting Zhang - Department of Mathematics, Zhejiang University, Hangzhou 310027, China (email)

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.

Received: May 2010;      Revised: September 2010;      Published: December 2010.

 References