# High Mach number limit of one-dimensional piston problem for non-isentropic compressible Euler equations: Polytropic gas

## TL;DR

This paper investigates the high Mach number limit of the one-dimensional piston problem for polytropic gases governed by the full compressible Euler equations, analyzing measure solutions and their convergence as Mach number tends to infinity.

## Contribution

It formulates the piston problem within the framework of Radon measure solutions and proves convergence of weak solutions to measure solutions in the high Mach number limit.

## Key findings

- Weak solutions converge to measure solutions as Mach number increases.
- The formulation handles mass concentration and vacuum formation.
- Convergence is established in the sense of measures.

## Abstract

We study high Mach number limit of the one dimensional piston problem for the full compressible Euler equations of polytropic gas, for both cases that the piston rushes into or recedes from the uniform still gas, at a constant speed. There are two different situations, and one needs to consider measure solutions of the Euler equations to deal with concentration of mass on the piston, or formation of vacuum. We formulate the piston problem in the framework of Radon measure solutions, and show its consistency by proving that the integral weak solutions of the piston problems converge weakly in the sense of measures to (singular) measure solutions of the limiting problems, as the Mach number of the piston increases to infinity.