On the one-dimensional piston model with Large Velocity Variations

TL;DR

This paper analyzes the dynamics of a one-dimensional piston expanding into a rarefied gas, deriving estimates and proving the global existence of solutions with significant velocity variations in the vanishing-density limit.

Contribution

It provides new asymptotic estimates and a proof of global-in-time existence for solutions with large velocity variations in the piston model.

Findings

Derived sharp estimates for piston-shock distance and wave interactions

Proved global-in-time existence of piecewise smooth solutions

Identified a stable mechanism in the vanishing-density limit

Abstract

This paper investigates the dynamics of a one-dimensional piston expanding into a static rarefied gas. Using asymptotic analysis in the limit of vanishing initial density, we derive sharp estimates for the piston-shock distance, the separation of characteristic speeds, and the reflection coefficient associated with characteristic waves interacting with the leading shock front. Based on these estimates, we apply the method of characteristics to prove the global-in-time existence of piecewise smooth solutions. The resulting flow structure exhibits significant velocity variations. The analysis reveals a stable mechanism that operates in the vanishing-density limit of the piston model.