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Rescaled objective solutions of Fokker-Planck and Boltzmann equations
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Matthies, Karsten and Theil, Florian (2019) Rescaled objective solutions of Fokker-Planck and Boltzmann equations. SIAM Journal of Mathematical Analysis . ISSN 0036-1410.
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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: |
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| 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 | ||||
| Date of first compliant deposit: | 11 February 2019 | ||||
| Date of first compliant Open Access: | 11 February 2019 | ||||
| Related URLs: |
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