Stability of Solutions of Hydrodynamic Equations Describing the Scaling Limit of a Massive Piston in an Ideal gas

TL;DR

This paper investigates the stability of solutions to a hydrodynamic equation modeling a massive piston in an ideal gas, providing explicit stability criteria and analyzing the behavior near stable and unstable solutions.

Contribution

It derives explicit stability criteria for the hydrodynamic equation and explores the dynamics near stable and unstable stationary solutions.

Findings

Explicit criteria for global stability of solutions.

Existence of linearly unstable solutions for certain parameters.

Evidence of long-term stability near stable solutions in the mechanical system.

Abstract

We analyze the stability of stationary solutions of a singular Vlasov type hydrodynamic equation (HE). This equation was derived (under suitable assumptions) as the hydrodynamical scaling limit of the Hamiltonian evolution of a system consisting of a massive piston immersed in an ideal gas of point particles in a box. We find explicit criteria for global stability as well as a class of solutions which are linearly unstable for a dense set of parameter values. We present evidence (but no proof) that when the mechanical system has initial conditions ``close'' to stationary stable solutions of the HE then it stays close to these solutions for a time which is long compared to that for which the equations have been derived. On the other hand if the initial state of the particle system is close to an unstable stationary solutions of the HE the mechanical motion follows for an extended time a perturbed solution of that equation: we find such approximate periodic solutions that are linearly stable.