Global Existence Of Shock Solutions For One-dimensional Non-isentropic Piston Problem

In this paper,the existence of shock solutions for one-dimensional non-isentropic piston problem is studied by using the modified Glimm scheme when the initial value and the piston velocity are all constant small perturbations.Wang has studied the existence of shock solutions for one-dimensional isentropic piston problem in [37].Compared with reference [37],the main innovations of this paper are as follows:(1)in this paper,three equations are studied,and the second characteristic family is linear degeneracy,and the solution of the Riemann problem contains contact discontinuity;(2)the initial value in this paper is no longer a constant,but a small perturbation of the constant.In order to get the conclusion of this paper,firstly,the solvability of the Riemann piston problem is given.Secondly,the interaction of the wave and the reflection of the disturbance wave on the main shock wave and piston are estimated accurately.Then the Glimm functional is constructed and the monotonicity of the Glimm functional is proved according to the interaction of the wave and the estimation of the reflection of the perturbation wave on the main shock wave and the piston.Finally,the global existence of shock solution is obtained according to the frame of Glimm scheme without limiting the strength of the main shock wave.