Asymptotic behaviour for the Vlasov-Poisson system in the stellar-dynamics

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. Keywords. Vlasov-Poisson system – stellar dynamics – polytropic gas spheres – gravitation – mass – energy – kinetic energy – potential energy – interpolation – Hardy-Littlewood-Sobolev inequality – optimal constant – symmetric nonincreasing rearrangements – Riesz ’ theorem – bounded solutions – direct variational methods – minimization – scalings – solutions with compact support – Galilean invariance – pseudo-conformal law – variance – dispersio