  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.
