The Well-posedness For The Cauchy Problems Of Radiation Hydrodynamics Equations With Viscosity

In this paper we are interested in the well-posedness for the Cauchy problems ofthe Navier-Stokes and Boltzmann coupled equations in radiation hydrodynamics.Theimportance of thermal radiation in physical problems increases as the temperature israised. At moderate temperatures, the role of the radiation is primarily one of trans-porting energy by radiative process, while at higher temperature, the energy and mo-mentum densities of the radiation field may become comparable to or even dominatethe corresponding ?uid quantities. Since this kind of equation has important and wideapplications in practice, it has received many attentions, such as Xinhua Zhong andSong Jiang [1] studied the well-posedness for the Cauchy problems of the Euler andBoltzmann coupled equations in radiation hydrodynamics. This article is based onthe their results and studies the Cauchy problems of the Navier-Stokes and Boltzmanncoupled equations in radiation hydrodynamics. The paper mainly includes followingseveral aspects:In the first part, we introduce the main mathematical models of radiation hydro-dynamics and the origin of the model, and discuss some nature of amounts such asσsandσa .In the second part, we investigate the well-posedness of the Cauchy problems forthe equations of one-dimensional isothermal radiation hydrodynamics with viscosity. We obtain the local existence, uniqueness and stability of smooth solutions to thisproblem by using the energy method and the Picard iteration.In the third part, we consider the well-posedness of the Cauchy problems for theequations of multi-dimensional radiation hydrodynamics with viscosity . To concludethis result , we shall mainly use estimates of the Navier-Stokes equations in severalspace variables and the transport equations, and the iteration scheme.