Piston problem for the isentropic Euler equations for a modified Chaplygin gas

Abstract

We constructively solve the piston problem for the one-dimensional isentropic Euler equations for a modified Chaplygin gas. We give a rigorous proof of the global existence and uniqueness of a shock wave separating constant states ahead of the piston when the piston advances into the gas. The results are quite different from those for a pure Chaplygin gas or a generalized Chaplygin gas, in which a Radon measure solution is constructed to deal with the concentration of mass on the piston. When the piston recedes from the gas, we show strictly that only a first-family rarefaction wave exists in front of the piston and that concentration will never occur. In addition, by studying the limiting behavior, we show that the piston solutions of the modified Chaplygin gas equations tend to the piston solutions of the generalized or pure Chaplygin gas equations as a single parameter of the pressure state function vanishes.