Young Researchers Workshop: Kinetic models in biology and social sciences

Asymptotic behaviors of the Vlasov-Poisson-Fokker-Planck equation

Maxime Herda

University Pierre and Marie Curie

In this talk, I will present some results obtained in collaboration with L. Miguel Rodrigues (Univ. Rennes 1). We are interested in the two-dimensional Vlasov-Poisson-Fokker-Planck equation for charged particles in a steady inhomogeneous background of opposite charges. Our goal is to provide uniform global in time hypocoercive and regularisation estimates, by building one Lyapunov functional, where dependencies on the mean-free path and the Debye length are made explicit. In our analysis the mean free path covers the full range of possible values: from the regime of evanescent collisions to the strongly collisional regime. From these estimates, we build solutions, show exponential decay to equilibrium with explicit asymptotic rates in both regimes and derive asymptotic models in the strongly collisional regime.