Matthies, Karsten and Theil, Florian (2019) Rescaled objective solutions of Fokker-Planck and Boltzmann equations. SIAM Journal of Mathematical Analysis .

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Abstract

We study the long-time behavior of symmetric solutions of the nonlinear Boltzmann equation and a closely related nonlinear Fokker-Planck equation. If the symmetry of the solutions corresponds to shear flows, the existence of stationary solutions can be ruled out because the energy is not conserved. After anisotropic rescaling both equations conserve the energy. We show that the rescaled Boltzmann equation does not admit stationary densities of Maxwellian type (exponentially decaying). For the rescaled Fokker-Planck equation we demonstrate that all solutions converge to a Maxwellian in the long-time limit, however the convergence rate is only algebraic, not exponential.

Item Type:	Journal Article
Subjects:	Q Science > QC Physics
Divisions:	Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH):	Transport theory, Fokker-Planck equation
Journal or Publication Title:	SIAM Journal of Mathematical Analysis
Publisher:	Society for Industrial and Applied Mathematics
ISSN:	0036-1410
Official Date:	30 January 2019
Dates:	Date Event 30 January 2019 Accepted
Status:	Peer Reviewed
Publication Status:	Published
Reuse Statement (publisher, data, author rights):	“First Published in SIAM Journal of Mathematical Analysis in [volume and number, or year], published by the Society for Industrial and Applied Mathematics (SIAM) ”Unauthorized reproduction of this article is prohibited.”
Access rights to Published version:	Restricted or Subscription Access
Copyright Holders:	SIAM
Related URLs:	Publisher

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