Existence results and blow-up criterion of compressible radiation hydrodynamic equations

TL;DR

This paper proves the existence of unique local strong solutions for 3D compressible radiation hydrodynamic equations with large initial data, including vacuum regions, and establishes a blow-up criterion based on key variables.

Contribution

It introduces new existence results for solutions with initial vacuum and irregular domains, and provides a blow-up criterion for the equations.

Findings

Existence of unique local strong solutions with large initial data.

Initial vacuum regions are allowed without bounded density away from zero.

A Beal-Kato-Majda type blow-up criterion is established.

Abstract

In this paper, we consider the $3$D compressible radiation hydrodynamic (RHD) equations with thermal conductivity in a bounded domain. The existence of unique local strong solutions is firstly established when the initial data are arbitrarily large and satisfy some initial layer compatibility condition. The initial mass density needs not be bounded away from zero and may vanish in some open set. Moreover, we show that if the initial vacuum domain is not so irregular, then the compatibility condition is necessary and sufficient to guarantee the existence of the unique strong solution. Finally, we show a Beal-Kato-Majda type blow-up criterion in terms of $(\nabla I,\rho,\theta)$.