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

2015/09/11 by Hwang, Hyung Ju, Jang, Juhi, Velázquez, Juan J. L.
#35A20 #35K65 #35Q84 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1509.03366

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

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