American Institute of Mathematical Sciences

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March  2009, 2(1): 163-192. doi: 10.3934/dcdss.2009.2.163

Interior regularity of the compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum

Yuming Qin 1, , Lan Huang 2, , Shuxian Deng 2, , Zhiyong Ma 2, , Xiaoke Su 2, and  Xinguang Yang 2,

1. 

Department of Mathematics - Henan University,Kaifeng 475001
2. 

College of Information Science and Technology, Donghua University, Shanghai 201620, China, China, China, China, China

Received  July 2008 Revised  November 2008 Published  January 2009

This paper is concerned with the interior regularity of global solutions for the one-dimensional compressible isentropic Navier-Stokes equations with degenerate viscosity coefficient and vacuum. The viscosity coefficient $\mu$ is proportional to $\rho^{\theta}$ with $0<\theta<1/3$, where $\rho$ is the density. The global existence has been established in [44] (Vong, Yang and Zhu, J. Differential Equations, 192(2), 475--501). Some ideas and more delicate estimates are introduced to prove these results.
Keywords: regularity, degeneration, viscosity, vacuum., Compressible Navier-Stokes equations.
Mathematics Subject Classification: Primary: 35Q30, 35D10; Secondary: 76N1.
Citation: Yuming Qin, Lan Huang, Shuxian Deng, Zhiyong Ma, Xiaoke Su, Xinguang Yang. Interior regularity of the compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum. Discrete and Continuous Dynamical Systems - S, 2009, 2 (1) : 163-192. doi: 10.3934/dcdss.2009.2.163
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