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

Hwang, Hyung Ju; Jang, Juhi; Velázquez, Juan

doi:10.1090/qam/1507

Authors: Hyung Ju Hwang, Juhi Jang and Juan J. L. Velázquez
Journal: Quart. Appl. Math. 77 (2019), 19-70
MSC (2010): Primary 35Q84, 35K65, 35A20; Secondary 35Q70, 35R60, 35R06, 60H15, 60H30, 47D07
DOI: https://doi.org/10.1090/qam/1507
Published electronically: June 19, 2018
MathSciNet review: 3897919
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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 (Nonuniqueness for the kinetic–Fokker–Planck equation with inelastic boundary conditions) to prove rigorously nonuniqueness of solutions for the kinetic Fokker–Planck equation with inelastic boundary conditions.

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