Initial Value Problems For Isentropic Multi-pressure Euler System

The multi-pressure of Euler system can be regarded as a natural asymptotic system for the multi-pressure Navier-Stokes equations in the regime of infinite Reynolds number,and can be used to describe more complex turbulence models with different turbulent temperatures.This paper mainly studies the vanishing pressure limit of Riemann solutions and the global classical solution of the Cauchy problem for the one-dimensional isentropic multi-pressure Euler equations.In the first chapter,the research background and current situation of the isentropic Euler equation and multi-pressure Euler system are introduced.The second chapter reviews the Riemann problem and Riemann solution of the zero-pressure flow.In the third chapter,the Riemann solutions of the isentropic multi-pressure Euler system are constructed by using the phase plane analysis method,and the 4types of elementary wave solutions are obtained: forward rarefaction wave,backward rarefaction wave,forward shock wave and backward shock wave.It is then proved that,when the pressure limit vanishes,the two-shock solution of the isentropic multipressure Euler system tends to the delta shock solution of the zero-pressure flow,and the two-rarefaction-wave solution of the isentropic multi-pressure Euler system tends to the vacuum solution to the zero-pressure flow.In chapter four,we study Cauchy problem of the isotropic multi-pressure Euler system.Firstly,it is proved that when there is no vacuum at the initial moment,there will be no vacuum in the solution of the Cauchy problem.Secondly,using the characteristic analysis method and the uniform a priori estimate,by studying the global smooth solution of a kind of Goursat problem,it is proved that the Cauchy problem admits a unique global continuous and piecewise smooth solution under the condition of the initial value of continuous and piecewise smooth function.