Piston problem to the isentropic Euler equations for modified Chaplygin gas

TL;DR

This paper constructively solves the piston problem for the isentropic Euler equations of modified Chaplygin gas, proving global existence and uniqueness of shock waves and analyzing limiting behaviors compared to other Chaplygin gases.

Contribution

It provides a rigorous construction of piston solutions for modified Chaplygin gas, including shock wave existence, uniqueness, and limiting behavior analysis, differing from prior Radon measure solutions.

Findings

Global existence and uniqueness of shock waves when piston pushes into gas.

Only the first family rarefaction wave exists when piston pulls back.

Piston solutions tend to those of generalized or pure Chaplygin gas as a parameter vanishes.

Abstract

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