# American Institute of Mathematical Sciences

March  2011, 10(2): 459-478. doi: 10.3934/cpaa.2011.10.459

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

 1 Department of Mathematics, Zhejiang University, Hangzhou 310027, China

Received  May 2010 Revised  September 2010 Published  December 2010

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.
Citation: Ping Chen, Daoyuan Fang, Ting Zhang. Free boundary problem for compressible flows with density--dependent viscosity coefficients. Communications on Pure & Applied Analysis, 2011, 10 (2) : 459-478. doi: 10.3934/cpaa.2011.10.459
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