The Boltzmann equation without angular cutoff in the whole space : I, Global existence for soft potential

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

Related Research Unit(s)

Detail(s)

Original languageEnglish
Pages (from-to)915-1010
Journal / PublicationJournal of Functional Analysis
Volume262
Issue number3
Publication statusPublished - 1 Feb 2012

Abstract

It is known that the singularity in the non-cutoff cross-section of the Boltzmann equation leads to the gain of regularity and a possible gain of weight in the velocity variable. By defining and analyzing a non-isotropic norm which precisely captures the dissipation in the linearized collision operator, we first give a new and precise coercivity estimate for the non-cutoff Boltzmann equation for general physical cross-sections. Then the Cauchy problem for the Boltzmann equation is considered in the framework of small perturbation of an equilibrium state. In this part, for the soft potential case in the sense that there is no positive power gain of weight in the coercivity estimate on the linearized operator, we derive some new functional estimates on the nonlinear collision operator. Together with the coercivity estimates, we prove the global existence of classical solutions for the Boltzmann equation in weighted Sobolev spaces. © 2011 Elsevier Inc.

Research Area(s)

  • Boltzmann equation, Coercivity estimate, Global existence, Non-cutoff cross-sections, Non-isotropic norm, Soft potential