On the structure of the singular set for the kinetic Fokker-Planck equations in domains with boundaries

TL;DR

This paper analyzes the asymptotic behavior of solutions to the kinetic Fokker-Planck equation with inelastic boundaries, revealing nonuniqueness of solutions when the restitution coefficient is below a critical value, and explaining discrepancies in physical simulations.

Contribution

It provides asymptotic analysis of solutions near singular points, demonstrating nonuniqueness and connecting mathematical results with physical simulation behaviors.

Findings

Solutions are nonunique for r < r_c due to interactions at the singular point

Different solutions interact differently with a Dirac mass at (0,0)

Asymptotics explain varied behaviors in physics literature

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 [34] to prove rigorously nonuniqueness of solutions for the kinetic Fokker-Planck equation with inelastic boundary conditions.