Existence Results and Blow-Up Criterion of Compressible Radiation Hydrodynamic Equations

In this paper, we consider the 3D compressible radiation hydrodynamic equations with thermal conductivity in a bounded domain. The existence of unique local strong solutions with vacuum is firstly established when the initial data are arbitrarily large and satisfy some initial layer compatibility condition. 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, a Beale–Kato–Majda type blow-up criterion is shown in terms of (∇I,ρ,θ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\nabla I,\rho ,\theta )$$\end{document}.