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arXiv · 1509.03366

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

Hyung Ju Hwang·Juhi Jang·Juan J. L. Velázquez
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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].

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Hyung Ju Hwang, Juhi Jang, Juan J. L. Velázquez. 2015-09-11. Nonuniqueness for the kinetic Fokker-Planck equation with inelastic boundary conditions. https://arxiv.org/abs/1509.03366

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