The Fokker-Planck equation for bosons in 2D: well-possedness and asymptotic analysis

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.

References in corpus (2)

• A conservative spectral method for the Boltzmann equation with anisotropic scattering and the grazing collisions limit
• On mathematical models for Bose-Einstein condensates in optical lattices (expanded version)

Cited by in corpus (5)

• On the singularity formation and relaxation to equilibrium in 1D Fokker-Planck model with superlinear drift
• Numerical study of Bose-Einstein condensation in the Kaniadakis-Quarati model for bosons
• Singularities in $L^1$-supercritical Fokker-Planck equations: A qualitative analysis
• Convergence of a finite volume scheme and dissipative measure-valued-strong stability for a hyperbolic-parabolic cross-diffusion system
• Local well-posedness of a nonlinear Fokker-Planck model