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

.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 a 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 as 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 a one-dimensional periodic box \(\mathbb{T}\cong \mathbb{R}/\mathbb{Z}\) with viscosity and thermal conductivity does exist as 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 \(L^m([0,T], L^\infty (\mathbb{T}))\text{-}\)estimate on the absolute temperature for some \(m\in \mathbb{N}\) 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.Keywordsradiation hydrodynamics model with viscosity and thermal conductivityglobal large solutionsdissipative estimates on the first-order spatial derivatives of the bulk velocity and the absolute temperaturepointwise estimatesMSC codes76N1576N1735B4035M3135Q35