Nonuniqueness for the Kinetic Fokker–Planck Equation with Inelastic Boundary Conditions

Abstract

We describe the structure of solutions of the kinetic Fokker–Planck equations in domains with boundaries near the singular set in one-space dimension. We study in particular the behaviour of the solutions of this equation for inelastic boundary conditions which are characterized by means of a coefficient r describing the amount of energy lost in the collisions of the particles with the boundaries of the domain. A peculiar feature of this problem is the onset of a critical exponent rc which follows from the analysis of McKean (J Math Kyoto Univ 2:227–235 1963) of the properties of the stochastic process associated to the Fokker–Planck equation under consideration. In this paper, we prove rigorously that the solutions of the considered problem are nonunique if r < rc and unique if $${r_{c} < r \leqq 1.}$$ In particular, this nonuniqueness explains the different behaviours found in the physics literature for numerical simulations of the stochastic differential equation associated to the Fokker–Planck equation. In the proof of the results of this paper we use several asymptotic formulas and computations in the companion paper (Hwang in Q Appl Math 2018).