The Well-posedness Of The Problems For The Radiation Hydrodynamics Equations

Some mathematical problems in radiation hydrodynamics equations are considered in this thesis, including the local existence of smooth solutions to the initial boundary value problem for the one-dimensional and multidimensional radiation hydrodynamics equations, the formation of singularities of solutions to the three-dimensional radiation hydrodynamics equations and the global weak solutions to the one-dimensional radiation hydrodynamics equations.The radiation hydrodynamics concerns the propagation of thermal radiation through a fluid, and the effect of this radiation on the hydrodynamics describing the fluid motion. The importance of thermal radiation in physical problems increases as the temperature is raised, because the radiation energy density varies as the fourth power of the temperature. At mod-erate temperatures (say, thousands of degrees Kelvin), the role of the radiation is transporting energy by radiative processes. At higher temperatures (say, millions of degrees Kelvin), the energy and momentum densities of the radiation field may become dominating the corre-sponding fluid quantities. In this case, the radiation field significantly affects the dynamics of the fluid. Therefore, at higher temperature condition, one must consider the functions of radiation when we study the motion of fluid. The theory of radiation hydrodynamics owns a wide range of application, including in astrophysical, laser fusion, supernove explosions.For one-dimensional and multidimensional radiation hydrodynamics equations, we first study the well-posedness of the initial-boundary value problems. By using the Picard iter-ation, energy estimate we obtain the local existence of smooth solutions. Second for the three-dimensional isentropic radiation hydrodynamics equations, we prove that some C1 so-lutions should blow-up in a finite time regardless of the size of the initial disturbance. Finally, for the Cauchy problem of the one-dimensional radiation hydrodynamics equations, by us-ing the compensated compactness argument, we establish the existence of a global entropy solution in L∞with arbitrarily large initial data.