Global existence and optimal decay rate of the classical solution to 3-D Radiative Hydrodynamics with and without Heat Conductivity

TL;DR

This paper proves the global existence and decay rates of classical solutions to 3-D radiative hydrodynamics equations, showing radiation can replace heat conduction in ensuring solution stability and decay.

Contribution

It establishes the existence, uniqueness, and decay properties of solutions with small initial perturbations, including cases without heat conductivity, highlighting radiation's role.

Findings

Solutions exist globally for small initial perturbations.

Solutions decay at a rate of (1+t)^(-3/4 - i/2) in certain norms.

Radiation can substitute heat conduction in maintaining solution decay.

Abstract

The classical solution of the 3-D radiative hydrodynamics model is studied in $H^k$-norm under two different conditions, with and without heat conductivity. We have proved the following results in both cases. First, when the $H^k$ norm of the initial perturbation around a constant state is sufficiently small and the integer $k\geq2$, a unique classical solution to such Cauchy problem is shown to exist. Second, if we further assume that the $L^1$ norm of the initial perturbation is small too, the i-order($0\leq i\leq k-2$) derivative of the solutions have the decay rate of $(1+t)^{-\frac 34-\frac i2}$ in $H^2$ norm. Third, from the results above we can see that for radiative hydrodynamics, the radiation can do the same job as the heat conduction, which means if the thermal conductivity coefficient turns to $0$, because of the effect of radiation, the solvability of the system and decay rate of the solution stay the same.