Nonuniqueness for the kinetic Fokker-Planck equation with inelastic boundary conditions

TL;DR

This paper investigates the nonuniqueness of solutions to the kinetic Fokker-Planck equation with inelastic boundary conditions, revealing a critical energy loss coefficient that determines solution uniqueness and explaining discrepancies in numerical simulations.

Contribution

It rigorously establishes the conditions for nonuniqueness and uniqueness of solutions based on the inelastic boundary coefficient, linking mathematical analysis with physical observations.

Findings

Solutions are nonunique if r < r_c.

Solutions are unique if r_c < r ≤ 1.

Nonuniqueness explains differing numerical results in physics literature.

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 (cf. [26]) 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 $r < r_c$ and unique if $r_{c}<r\leq 1$. 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 [16].