Exterior Problem For A Compressible Viscous And Heat-conductive Gas With Radiation

The dynamics of compressible viscous and heat-conductive gas with radiation can be described by a compressible Navier-Stokes type equations.The mathematical theories on the global well-posedness to such a system and the precise description of the large time behaviors of its global solutions is a hot topic in the field of nonlinear partial differential equations recently.The main purpose of this thesis is devoted to the global solvability and the large time behavior of global solutions to the exterior problem of such a system with large data which satisfies certain symmetric conditions.Our results show that the exterior problem to such a system admits a unique global strong solution and the unique global solution is also time-asymptotically stable,provided that the adiabatic exponent of the gas is sufficiently close to 1.The results contains both the spherically symmetric case and the cylindrically symmetric case and key point to yield the desired results lies in how to deduce the uniformly positive lower and upper bounds on both specific volume and the absolute temperature.The main steps in our analysis contains the following two parts:· The first step is to derive a decent localized version of the expression for the specific volume.From which,the basic energy type estimate for the energy and the a priori assumption that the absolute temperature is bounded by some uniform positive constants from both below and above,we can deduce the desired uniformly positive lower and upper bounds on the specific volume;· The second step is to deduce certain energy type estimates on the absolute temperature first,and then to close the a priori assumption imposed on the absolute temperature in the first step by these estimates and the assumption that the adiabatic exponent of the gas is sufficiently close to 1.By combining the well-established local solvability result to the exterior problem,the above a priori estimates and the standard continuation argument,we can indeed get our results.