Mathematical study of gravitational kinetic equations by considering relativistic effects : stability, self-similar solutions.
This document is concerned with the behavior of solutions near ground states for gravitational kinetic systems of Vlasov type. In the first chapter we build by variational methods some stationary states of the Vlasov-Manev system and we prove their orbital stability. The second chapter gives the existence of self-similar blow-up solutions to the "pur Vlasov-Manev" system near ground states. In the third chapter we obtain the orbital stability of a large class of ground states. New methods based on the rigidity of the flow are developed in these three chapters. In particular, they provide the uniqueness of ground states by avoiding the study of non-local Euler-Lagrange equations, they solve a variational problem with non finite constraints and they give the orbital stability of ground states which are not necessary obtained from variational methods. In the fourth chapter, we finish our analysis with a numerical study of the radialy symmetric Vlasov-Poisson system : we give numerical finite difference schemes which conserve the mass and the Hamiltonien of the system.