Inventiones mathematicae

, Volume 207, Issue 1, pp 115–290

Regularity of the Boltzmann equation in convex domains

  • Yan Guo
  • Chanwoo Kim
  • Daniela Tonon
  • Ariane Trescases
Article

Abstract

A basic question about regularity of Boltzmann solutions in the presence of physical boundary conditions has been open due to characteristic nature of the boundary as well as the non-local mixing of the collision operator. Consider the Boltzmann equation in a strictly convex domain with the specular, bounce-back and diffuse boundary condition. With the aid of a distance function toward the grazing set, we construct weighted classical \(C^{1}\) solutions away from the grazing set for all boundary conditions. For the diffuse boundary condition, we construct \(W^{1,p}\) solutions for \(1< p<2\) and weighted \( W^{1,p}\) solutions for \(2\le p\le \infty \) as well.

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Yan Guo
    • 1
  • Chanwoo Kim
    • 2
  • Daniela Tonon
    • 3
  • Ariane Trescases
    • 4
  1. 1.Division of Applied MathematicsBrown UniversityProvidenceUSA
  2. 2.Department of MathematicsThe University of Wisconsin-MadisonMadisonUSA
  3. 3.CEREMADE (UMR CNRS 7534), Université Paris-Dauphine, PSL Research UniversityParis Cedex 16France
  4. 4.CMLACachanFrance

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