Rescaled Objective Solutions of Fokker-Planck and Boltzmann equations

Karsten Matthies, Florian Theil

Research output: Contribution to journalArticle

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.
LanguageEnglish
Pages1321–1348
Number of pages28
JournalSIAM Journal on Mathematical Analysis (SIMA)
Volume51
Issue number2
DOIs
StatusPublished - 18 Apr 2019

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Fokker-Planck equation
shear flow
energy
symmetry

Keywords

  • Boltzmann equation
  • Fokker-Planck
  • Hypocoercivity
  • Objective solution

ASJC Scopus subject areas

  • Analysis
  • Computational Mathematics
  • Applied Mathematics

Cite this

Rescaled Objective Solutions of Fokker-Planck and Boltzmann equations. / Matthies, Karsten; Theil, Florian.

In: SIAM Journal on Mathematical Analysis (SIMA), Vol. 51, No. 2, 18.04.2019, p. 1321–1348.

Research output: Contribution to journalArticle

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