Existence Of A Shock Wave In A One-dimensional Radiation Hydrodynamic System

At high temperature, radiation has significant influence on the motion of fluids. Thus, the study of the mathematical theory for the radiation hydrodynamic systems is important not only in academic meaning, but also in applications. In recent years, there are some results in the theory of smooth solutions to the Cauchy problem for radiation hydrodynamic systems; while there is almost no any result on discontinuous solutions to this problem, which is more interesting from the view of applications.In this paper, we mainly study shock waves in a one-dimensional radiation hydrodynamic system. By using the Rankine-Hugoniot condition and entropy condition, this problem can be formulated as an initial boundary problem with a free boundary for radiation hydrodynamic system. First, we transform this free boundary to the fixed one by using change of variables involving unknowns. Then we investigate the existence and uniqueness of the solution to the initial boundary problem for this nonlinear system. For this problem, we first construct an approximate solution by using the compatibility conditions of the data. Then we use the Picard iteration and the Newton iteration for this nonlinear system respectively to construct a sequence of approximate solutions. By using a series of estimates and a compactness argument, we get the convergence of the sequence of approximate solutions. The limit of this sequence gives a shock wave of the original radiation hydrodynamic system.