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March  2008, 7(2): 373-381. doi: 10.3934/cpaa.2008.7.373

One dimensional compressible Navier-Stokes equations with density-dependent viscosity and free boundaries

Xulong Qin 1, , Zheng-An Yao 2, and  Hongxing Zhao 3,

1. 

Department of Mathematics, Sun Yat-Sen University, Guangzhou, 510275, China
2. 

Department of Mathematics, Sun Yat-sen University, Guangzhou 510275
3. 

Department of Mathematics, College of Mathematics and Information Science, GuangZhou University, Guangzhou, 510006, China

Received  July 2006 Revised  December 2007 Published  December 2007

A free-boundary problem is studied for the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity that decreases (to zero) with decreasing density, i.e., $\mu=A\rho^\theta$, where $A$ and $\theta$ are positive constants. The existence and uniqueness of the global weak solutions are obtained with $\theta\in (0,1]$, which improves the previous results and no vacuum is developed in the solutions in a finite time provided the initial data does not contain vacuum.
Keywords: Compressible Navier-Stokes equations, existence, uniqueness, density-dependent viscosity, free boundary..
Mathematics Subject Classification: Primary: 35Q30, 76N10; Secondary: 76N1.
Citation: Xulong Qin, Zheng-An Yao, Hongxing Zhao. One dimensional compressible Navier-Stokes equations with density-dependent viscosity and free boundaries. Communications on Pure and Applied Analysis, 2008, 7 (2) : 373-381. doi: 10.3934/cpaa.2008.7.373
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