Linear stability of stationary solutions of the Vlasov-Poisson system in three dimensions

Rigorous results on the stability of stationary solutions of the Vlasov-Poisson system are obtained in the contexts of both plasma physics and stellar dynamics. It is proved that stationary solutions in the plasma physics (stellar dynamics) case are linearly stable if they are decreasing (increasing) functions of the local, i.e., particle, energy. The main tool in the analysis is the free energy, a conserved quantity of the linearized system. In addition, an appropriate global existence result is proved for the linearized Vlasov-Poisson system and the existence of stationary solutions which satisfy the above stability condition is established.