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January  1998, 4(1): 1-32. doi: 10.3934/dcds.1998.4.1

Vacuum states for compressible flow

Tai-Ping Liu 1, , Zhouping Xin 2, and  Tong Yang 3,

1. 

Department of Mathematics, Stanford University, United States
2. 

Department of Mathematics, Courant Institute, New York University, United States
3. 

Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

Received  June 1997 Published  October 1997

In this paper we study the evolutions of the interfaces between gases and the vacuum for both inviscid and viscous one dimensional isentropic gas motions. The local (in time) existence of solutions for both inviscid and viscous models with initial data containing vacuum states is proved and some singular properties on the free surfaces separating the gas and the vacuum are obtained. It is found that the Euler equations are better behaved near the vacuum than the compressible Navier-Stokes equations. The Navier-Stokes equations with viscosity depending on density are introduced, which is shown to be well-posed (at least locally) and yield the desired solutions near vacuum.
Keywords: Compressible flow, Navier-Stokes equations., Euler equations.
Mathematics Subject Classification: 35Qx.
Citation: Tai-Ping Liu, Zhouping Xin, Tong Yang. Vacuum states for compressible flow. Discrete and Continuous Dynamical Systems, 1998, 4 (1) : 1-32. doi: 10.3934/dcds.1998.4.1
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