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Nonuniqueness for the kinetic Fokker-Planck equation with inelastic boundary conditions. (English) Zbl 1408.35190

Summary: 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 \(r_{c}\) which follows from the analysis of H. P. McKean jun. [J. Math. Kyoto Univ. 2, 227–235 (1963; Zbl 0119.34701)] 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 \({r_{c} < r \leqq 1.}\) 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 our companion paper [Q. Appl. Math. 77, No. 1, 19–70 (2019; Zbl 1404.35446)].

MSC:

35Q84 Fokker-Planck equations
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
35B33 Critical exponents in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
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