Global Regularity for A Radiation Hydrodynamics Model with Viscosity and Thermal Conductivity

TL;DR

This paper proves the global existence and uniqueness of smooth solutions for a one-dimensional radiation hydrodynamics model with viscosity and thermal conductivity, for any size of initial data, by establishing delicate a priori estimates.

Contribution

It demonstrates that the combined effects of viscosity and thermal conductivity ensure global regularity for large initial data in a radiation hydrodynamics model, extending previous results.

Findings

Global smooth solutions exist for all large initial data.

Delicate estimates on temperature and flux are crucial for analysis.

Positive bounds on density and temperature are established.

Abstract

In this paper, we study the global wellposedness of a radiation hydrodynamics model with viscosity and thermal conductivity. It is now well-understood that, unlike the compressible Euler equations whose smooth solutions must blow up in finite time no matter how small and how smooth the initial data is, the dissipative structure of such a radiation hydrodynamics model can indeed guarantee that its one-dimensional Cauchy problem admits a unique global smooth solution provided that the initial data is sufficiently small, while for large initial data, even if the heat conductivity is taken into account but the viscosity effect is ignored, shock type singularities must appear in finite time for smooth solutions of the Cauchy problem of one-dimensional radiation hydrodynamics model with thermal conductivity and zero viscosity. Thus a natural question is, if effects of both the viscosity and the thermal conductivity are considered, does the one-dimensional radiation hydrodynamics model with viscosity and thermal conductivity exist a unique global large solution? We give an affirmative answer to this problem and show in this paper that the initial-boundary value problem to the radiation hydrodynamics model in an one-dimensional periodic box T = R/Z with viscosity and thermal conductivity does exist a unique global smooth solution for any large initial data. The main ingredient in our analysis is to introduce some delicate estimates, especially an improved estimate on the absolute temperature and a pointwise estimate between the absolute temperature, the specific volume, and the first-order spatial derivative of the macro radiation flux, to deduce the desired positive lower and upper bounds on the density and the absolute temperature.