Piston problem for the modified Chaplygin Euler equations

Abstract

The piston problem for one-dimensional modified Chaplygin Euler equations and the limiting behavior of piston solutions are constructively solved when the piston rushes into or recedes from a modified Chaplygin gas. For the proceeding piston problem, we establish the global existence of a shock solution without restriction on the Mach number. Moreover, we study the high Mach number limiting behavior of the shock solution and clarify that the concentration of mass on the piston appears as $M_0\to \infty$. The results are different from the previous results for a pure Chaplygin gas or a generalized Chaplygin gas. For the receding case, we prove rigorously that only the first-family rarefaction wave exists in front of the piston and the concentration will never occur. Finally, it is proved that the limits of the piston solutions of the modified Chaplygin Euler equations are similar to the compressible Euler system with a pure Chaplygin gas as the pressure tends to the Chaplygin gas.