Global classical solutions to a multidimensional radiation hydrodynamics model with symmetry and large initial data

Abstract

As a first stage to study the global large solutions of the radiation hydrodynamics model with viscosity and thermal conductivity in the high-dimensional space, we study the problems in high dimensions with some symmetry, such as the spherically or cylindrically symmetric solutions. Specifically, we will study the global classical large solutions to the radiation hydrodynamics model with spherically or cylindrically symmetric initial data. The key point is to obtain the strict positive lower and upper bounds of the density $\rho$ and the lower bound of the temperature $\theta$. Compared with the Navier–Stokes equations, these estimates in the present paper are more complicated due to the influence of the radiation. To overcome the difficulties caused by the radiation, we construct a pointwise estimate between the radiative heat flux $q$ and the temperature $\theta$ by studying the boundary value problem of the corresponding ordinary differential equation. And we consider a general heat conductivity: $\kappa (\rho ,\theta )⩾C(1+{\theta }^{\beta })$ if $\rho ⩽{\rho }_{+}$; $\kappa (\rho ,\theta )⩽C(1+{\theta }^{\beta })$ if $\rho ⩾{\rho }_{-}>0$. This can be viewed as the first result about the global classical large solutions of the radiation hydrodynamics model with some symmetry in the high-dimensional space.