Existence results for  compressible  radiation hydrodynamic equations with vacuum

Yachun Li; Shengguo Zhu

doi:10.3934/cpaa.2015.14.1023

Article Contents

2015, Volume 14, Issue 3: 1023-1052. Doi: 10.3934/cpaa.2015.14.1023

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Existence results for compressible radiation hydrodynamic equations with vacuum

Yachun Li^1, and
Shengguo Zhu^2,

1.
Department of Mathematics and Key Lab of Scientific and Engineering Computing (MOE), Shanghai Jiao Tong University, Shanghai 200240
2.
Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240

Received: August 31, 2014

Revised: December 31, 2014

Published: April 2015

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Abstract

In this paper, we consider the three-dimensional compressible isentropic radiation hydrodynamic (RHD) equations. The existence of unique local strong solutions is firstly proved when the initial data are arbitrarily large, contain vacuum and satisfy some initial layer compatibility condition. The initial mass density does not need to be bounded away from zero and may vanish in some open set. We also prove that if the initial vacuum is not so irregular, then the initial layer compatibility condition is necessary and sufficient to guarantee the existence of a unique strong solution. Finally, we establish a blow-up criterion for the strong solution that we obtained. The similar results also hold for the barotropic flow with general pressure law $p_m=p_m(\rho)\in C^1(\mathbb{\overline{R}}^+)$.

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Mathematics Subject Classification: Primary: 35Q35, 35D35; Secondary: 35A01, 35E15.

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