Uncertainty principle and kinetic equations

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journal

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Author(s)

  • R. Alexandre
  • Y. Morimoto
  • S. Ukai
  • C. J. Xu
  • T. Yang

Detail(s)

Original languageEnglish
Pages (from-to)2013-2066
Journal / PublicationJournal of Functional Analysis
Volume255
Issue number8
Publication statusPublished - 15 Oct 2008

Abstract

A large number of mathematical studies on the Boltzmann equation are based on the Grad's angular cutoff assumption. However, for particle interaction with inverse power law potentials, the associated cross-sections have a non-integrable singularity corresponding to the grazing collisions. Smoothing properties of solutions are then expected. On the other hand, the uncertainty principle, established by Heisenberg in 1927, has been developed so far in various situations, and it has been applied to the study of the existence and smoothness of solutions to partial differential equations. This paper is the first one to apply this celebrated principle to the study of the singularity in the cross-sections for kinetic equations. Precisely, we will first prove a generalized version of the uncertainty principle and then apply it to justify rigorously the smoothing properties of solutions to some kinetic equations. In particular, we give some estimates on the regularity of solutions in Sobolev spaces w.r.t. all variables for both linearized and nonlinear space inhomogeneous Boltzmann equations without angular cutoff, as well as the linearized space inhomogeneous Landau equation. © 2008 Elsevier Inc. All rights reserved.

Research Area(s)

  • Boltzmann equations, Kinetic equations, Landau equation, Microlocal analysis, Non-cutoff cross-sections, Uncertainty principle

Citation Format(s)

Uncertainty principle and kinetic equations. / Alexandre, R.; Morimoto, Y.; Ukai, S.; Xu, C. J.; Yang, T.

In: Journal of Functional Analysis, Vol. 255, No. 8, 15.10.2008, p. 2013-2066.

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journal