Asymptotic Behaviour for the Vlasov-Poisson System in the Stellar-Dynamics Case

We study an optimal inequality which relates potential and kinetic energies in an appropriate framework for bounded solutions of the Vlasov-Poisson (VP) system. Op- timal distribution functions, which are completely characterized, minimize the total energy. From this variational approach, we deduce bounds for the kinetic and poten- tial energies in terms of conserved quantities (mass and total energy) of the solutions of the VP system and a nonlinear stability result. Then we apply our estimates to the study of the large time asymptotics and observe two different regimes.