Measure solutions to piston problem for compressible fluid flow of generalized chaplygin gas

TL;DR

This paper investigates the piston problem for compressible flow of generalized Chaplygin gas, establishing existence of shock or measure solutions depending on Mach number and analyzing vacuum states and solution convergence.

Contribution

It introduces a novel analysis of piston problem solutions for generalized Chaplygin gas, including shock and measure solutions based on Mach number thresholds.

Findings

Existence of shock solutions for Mach numbers below critical value.

Presence of measure solutions with Dirac delta support at the piston for high Mach numbers.

Analysis of vacuum states and solution convergence as Mach number increases.

Abstract

We study the piston problem of the compressible fluid flow with the generalized Chaplygin gas. Depending on the inferential critical value of Mach number, we prove that, there exists an integral weak solution for the proceeding piston problem, consisting of a shock separating constant states ahead of the piston if Mach numbers less than this critical value, while a singular measure solution, with density containing a Dirac measure supported on the piston, shall be proposed to solve the proceeding piston problem if Mach numbers greater than or equal to the critical value. For the receding piston problem, rarefaction wave solution always exists when the piston recedes from the gas with any constant speed. Moreover, the occurrence of vacuum state and the convergence of solutions, as well as degeneration of equations are analyzed in the receding case as Mach number tends to infinity.