# Measure solutions of one-dimensional piston problem for compressible Euler equations of Chaplygin gas

## TL;DR

This paper investigates the solutions of the one-dimensional piston problem for the Euler equations of Chaplygin gas, revealing different solution types depending on piston speed and connecting to polytropic gas behavior at high Mach numbers.

## Contribution

It introduces integral weak solutions and singular measure solutions for various piston speeds, extending understanding of Chaplygin gas dynamics in piston problems.

## Key findings

- Existence of integral weak solutions for subsonic piston speeds.
- Introduction of singular measure solutions for sonic and supersonic speeds.
- As Mach number approaches infinity, solutions converge to those of polytropic gases.

## Abstract

We are concerned with the one-dimensional piston problem for the compressible Euler equations of Chaplygin gas. If the piston moves at constant subsonic speed to the uniform gas, there exists an integral weak solution for the piston problem, consisting of a shock separating constant states ahead of the piston. While if the speed of the piston is sonic or supersonic, a singular measure solution, with density containing a Dirac measure supported on the piston, shall be introduced to solve the problem. Integral weak solution exists for the piston receding from the gas with any constant speed, and there is no vacuum. In the extreme case as the Mach number of the piston goes to infinity, the limiting equations and solutions are the same as that for the polytropic gases.