Existence results for compressible radiation hydrodynamics equations with vacuum

TL;DR

This paper proves the local existence and uniqueness of strong solutions for 3-D compressible radiation hydrodynamics equations with vacuum, including conditions for blow-up and generalization to barotropic flows.

Contribution

It establishes the first local existence results for strong solutions with vacuum in 3-D compressible RHD equations, including necessary and sufficient conditions and blow-up criteria.

Findings

Existence of strong solutions with vacuum for large initial data

Necessary and sufficient conditions for solution existence

Blow-up criteria for local strong solutions

Abstract

In this paper, we consider the 3-D compressible isentropic radiation hydrodynamics (RHD) equations. The local existence of strong solutions with vacuum is firstly established when the initial data is arbitrarily large, contains vacuum and satisfy some initial layer compatibility condition. The initial mass density needs not be bounded away from zero, it may vanish in some open set or decay at infinity. We also prove that if the initial vacuum is not so irregular, then the compatibility condition of the initial data is necessary and sufficient to guarantee the existence of a unique strong solution. Finally, we prove a blow-up criterion for the local strong solution. The similar result also holds for the general barotropic flow with pressure law $p_m=p_m(\rho)\in C^1(\mathbb{\overline{R}}^+)$.