Existence Of Weak Solutions For Euler Equations Of Isentropic Relativity

The main work of this paper is to study the existence of large initial value weak solutions and periodic solutions of the isentropic relativistic Euler equations.This system is a relativistic version of the classical Euler equations,whose state equation is the equation of state satisfied by the isothermal ideal gas.The key method in this paper is to use the Glimm difference scheme.Firstly,we discuss the geometric properties of shock wave in Riemann invariant coordinate space,and obtain the existence and uniqueness of the solution of Riemann problem.Then we define the strength of shock wave and construct a linear function.We obtain the existence of large initial value solutions for the system of isentropic relativistic Euler equations.Finally,we consider the periodicity of initial data.According to Frid's study on the existence of periodic solutions of classical hydrodynamic equations in [22],and using an important conclusion in [53]: at any time,the average value of solutions in a period is the same as that of initial values in a period.We can obtain the existence of the entropy solution in BV space when the initial value and its total variation are bounded in a period.