CPAA
Global solutions of the free boundary problem for the compressible Navier-Stokes equations with density-dependent viscosity
Xulong Qin Zheng-An Yao
Communications on Pure & Applied Analysis 2010, 9(4): 1041-1052 doi: 10.3934/cpaa.2010.9.1041
A free boundary problem is investigated for viscous, compressible, heat-conducting, one-dimensional real gas with general large initial data. More precisely, the viscosity is assumed to be $\mu(\rho)=\rho^{\lambda}(\lambda>0)$, where $\rho$ is the density of the gas, and there is nonlinear dependence upon the density and temperature for the equations of state which are different from the linear dependence of perfect gas. It is also proved that no shock wave, vacuum, mass or heat concentration will be developed in a finite time and that the free boundary (interface) separating the gas and vacuum expands at a finite velocity.
keywords: density-dependent viscosity global existence. heat-conducting real gas initial boundary problem Viscous
CPAA
One dimensional compressible Navier-Stokes equations with density-dependent viscosity and free boundaries
Xulong Qin Zheng-An Yao Hongxing Zhao
Communications on Pure & Applied Analysis 2008, 7(2): 373-381 doi: 10.3934/cpaa.2008.7.373
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.

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