(1) Ceremade (UMR CNRS no. 7534), Université Paris IX-Dauphine, Place de Lattre de Tassigny, 75775 Paris Cédex 16, France;(2) Departamento de Matemática Aplicada, Facultad de ciencias, Universidad de Granada, 18071 Granada, Spain
Abstract:
We study an optimal inequality which relates potential and kinetic energies in an appropriate framework for bounded solutions of the Vlasov-Poisson (VP) system. Optimal distribution functions, which are completely characterized, minimize the total energy. From this variational approach, we deduce bounds for the kinetic and potential 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.
