American Institute of Mathematical Sciences

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July  2008, 7(4): 987-1016. doi: 10.3934/cpaa.2008.7.987

A vacuum problem for multidimensional compressible Navier-Stokes equations with degenerate viscosity coefficients

Ping Chen 1, and  Ting Zhang 1,

1. 

Department of Mathematics, Zhejiang University, Hangzhou 310027, China

Received  March 2007 Revised  December 2007 Published  April 2008

Local solutions of the multidimensional Navier-Stokes equations for isentropic compressible flow are constructed with spherically symmetric initial data between a solid core and a free boundary connected to a surrounding vacuum state. The viscosity coefficients $\lambda, \mu$ are proportional to $\rho^\theta$, $0<\theta<\gamma$, where $\rho$ is the density and $\gamma > 1$ is the physical constant of polytropic fluid. It is also proved that no vacuum develops between the solid core and the free boundary, and the free boundary expands with finite speed.
Keywords: Compressible Navier-Stokes equations, uniqueness., free boundary, density-dependent viscosity, vacuum, existence.
Mathematics Subject Classification: Primary: 35Q30; Secondary: 35R35, 76N1.
Citation: Ping Chen, Ting Zhang. A vacuum problem for multidimensional compressible Navier-Stokes equations with degenerate viscosity coefficients. Communications on Pure & Applied Analysis, 2008, 7 (4) : 987-1016. doi: 10.3934/cpaa.2008.7.987
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