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

Ping Chen; Ting Zhang

doi:10.3934/cpaa.2008.7.987

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2008, Volume 7, Issue 4: 987-1016. Doi: 10.3934/cpaa.2008.7.987

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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

Received: February 28, 2007

Revised: November 30, 2007

Published: June 2008

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Abstract

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.

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Mathematics Subject Classification: Primary: 35Q30; Secondary: 35R35, 76N10.

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