ON THE STRUCTURE OF THE SINGULAR SET FOR THE KINETIC FOKKER-PLANCK EQUATIONS IN DOMAINS WITH BOUNDARIES

Abstract

In this paper we compute asymptotics of solutions of the kinetic Fokker-Planck equation with inelastic boundary conditions which indicate that the solutions are nonunique if r < r(c). The nonuniqueness is due to the fact that different solutions can interact in a different manner with a Dirac mass which appears at the singular point (x, v) = (0, 0). In particular, this nonuniqueness explains the different behaviours found in the physics literature for numerical simulations of the stochastic differential equation associated to the kinetic Fokker-Planck equation. The asymptotics obtained in this paper will be used in a companion paper (Nonuniqueness for the kinetic-Fokker-Planck equation with inelastic boundary conditions) to prove rigorously nonuniqueness of solutions for the kinetic Fokker-Planck equation with inelastic boundary conditions.