Stability of solutions of hydrodynamic equations describing the scaling limit of a massive piston in an ideal gas

We analyse 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 that are linearly unstable for a dense set of parameter values. We present some numerical evidence that when the mechanical system (with a large number of particles) has initial conditions 'close' to stationary stable solutions of the HE, then it stays close to these solutions for a long time. On the other hand, if the initial state of the particle system is close to an unstable stationary solution of the HE, then the mechanical system diverges rapidly from that solution and later appears to develop long lasting periodic oscillations. We find similar (approximately periodic) solutions of the HE that are linearly stable.