The Fokker–Planck equation for bosons in 2D: Well-posedness and asymptotic behavior. (May 2016)

• Record Type: Journal Article
• Authors: Cañizo, José A.; Carrillo, José A.; Laurençot, Philippe; Rosado, Jesús
• Is Part Of: Nonlinear analysis. Volume 137(2016)
• Page Start: 291
• Page End: 305
• Publication Date: 2016-05
• Subjects: Bose–Einstein -- Entropy method -- Long-time asymptotics; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2015.07.030
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We show that solutions of the 2D Fokker–Planck equation for bosons are defined globally in time and converge to equilibrium, and this convergence is shown to be exponential for radially symmetric solutions. The main observation is that a variant of the Hopf–Cole transformation relates the 2D equation in radial coordinates to the usual linear Fokker–Planck equation. Hence, radially symmetric solutions can be computed analytically, and our results for general (non radially symmetric) solutions follow from comparison and entropy arguments. In order to show convergence to equilibrium we also prove a version of the Csiszár–Kullback inequality for the Bose–Einstein–Fokker–Planck entropy functional.