The Cauchy Problem For The Equations Of Radiation Hydrodynamics

In this thesis, we consider the global existence and uniqueness of solutions to the initialvalue problem for the equations of radiation hydrodynamics. Furthermore, we study thelarge time behaviors of the solutions. The thesis is arranged as follows:In Chapter 1, we review the physics background of the equations of radiation hydrody-namics and the history on studying the equations. We also introduce the problems addressedin this study and summarize the main results.In Chapter 2, we investigate a model system for the radiating gas, i.e. a hyperbolic-elliptic coupled system. In Section 1, we consider global existence and pointwise estimatesof solutions to the Cauchy problem for the model system of the radiating gas. First of all,by using the energy method, we obtain the global existence and uniqueness of the solutions.Then, we derive pointwise estimates of the Green function by studying the Green functionproblem. Moreover, by using the Duhamel principle and the pointwise estimates of theGreen function, we obtain pointwise estimates of the solutions when the initial perturbationcorresponding to a positive constant state is sufficiently small in Hs(Rn). Furthermore, weshow the optimal Lp-norm estimates of the solutions. In Section 2, we consider a modi-fied model system for the radiating gas. This model doesn't satisfy the Shizuta-Kawashimacondition. Here, we study the global existence and asymptotic behaviors of solutions to aregularity-loss type for nonlinear hyperbolic-elliptic system in multi-dimensions. By usinga weighted energy method, we obtain Lp-norm decay estimates of solutions to the Cauchyproblem. Fianlly, we get large time asymptotic behaviors of the solutions. In Section 3, westudy the global existence and pointwise estimates of solutions to an equation which satisfiesthe Shizuta-Kawashima condition.In Chapter 3, we consider the initial value problem for a radiation hydrodynamic model with viscosity in R3. In Section 1, we obtain the global existence and some decay estimatesof solutions to the problem. In order to prove a priori estimates of the solutions, we applyan energy method as that in [54]. In Section 2, under the proof of the global existenceof the solutions in Section 1, we obtain pointwise estimates of solutions. Here, the decayestimates of the Green function for the linearized system to the equations is used. In Section3, when omitting the radiation effect, we consider the local existence of weak solutions tothe isentropic spherically symmetric Navier-Stokes equations.In Chapter 4, we investigate the global existence and decay estimates of solutions tothe Cauchy problem for a model system of the radiating gas with large initial data. Firstly,we consider the blow up of classical solutions. Secondly, when the derivative of initial datasatisfies some smallness conditions, we obtain the global existence and decay estimates ofthe solutions by using an energy method.