On regular solutions to compressible radiation hydrodynamic equations with far field vacuum

Abstract

Abstract The Cauchy problem for three-dimensional (3D) isentropic compressible radiation hydrodynamic equations is considered. When both shear and bulk viscosity coefficients depend on the mass density ρ \rho in a power law ρ δ {\rho }^{\delta } (with 0 < δ < 1 0\lt \delta \lt 1 ), based on some elaborate analysis of this system’s intrinsic singular structures, we establish the local-in-time well-posedness of regular solution with arbitrarily large initial data and far field vacuum in some inhomogeneous Sobolev spaces by introducing some new variables and initial compatibility conditions. Note that due to the appearance of the vacuum, the momentum equations are degenerate both in the time evolution and viscous stress tensor, which, along with the strong coupling between the fluid and the radiation field, make the study on corresponding well-posedness challenging. For proving the existence, we first introduce an enlarged reformulated structure by considering some new variables, which can transfer the degeneracies of the radiation hydrodynamic equations to the possible singularities of some special source terms, and then carry out some singularly weighted energy estimates carefully designed for this reformulated system.