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

Abstract

Abstract. In this paper, we consider the 3D compressible radiation hydrodynamic (RHD) 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 Beal-Kato-Majda type blow-up criterion is shown in terms of (∇I, ρ, θ).

Blow-up criterion for the 3D viscous polytropic fluids with degenerate viscosities

“…We also refer readers to [3], [6], [10], [13], [18], [26] and references therein for other interesting progress for this compressible degenerate system, corresponding radiation hydrodynamic equations and magnetohydrodynamic equations. It should be noted that one should not always expect the global existence of solutions with better regularities or general initial data because of Xin's results [23] and Rozanova's results [20].…”

Local well‐posedness of compressible radiation hydrodynamic equations with density‐dependent viscosities and vacuum

“…Li-Zhu studied the formulation of singularities to classical solutions with compactly supported density 18 and established the local well-posedness of strong solution containing a vacuum in homogeneous Sobolev space for general initial data satisfying initial compatibility conditions. 19,20 They also investigated the existence and uniqueness of local regular solutions for Euler-Boltzmann equations. 21 We mention that the above results are all for the case of constant viscosity coefficients.…”