Compactness in kinetic transport equations and hypoellipticity

Diogo Arsénio, Laure Saint-Raymond

Research output: Contribution to journalArticle

Abstract

We establish improved hypoelliptic estimates on the solutions of kinetic transport equations, using a suitable decomposition of the phase space. Our main result shows that the relative compactness in all variables of a bounded family of nonnegative functions fλ(x,v)∈L1 satisfying some appropriate transport relation. ν·∇xfλ=(1-Δx )β2(1-Δν)α2gλ may be inferred solely from additional integrability and compactness with respect to v. In a forthcoming work, the authors make a crucial application of this new approach to the study of the hydrodynamic limit of the Boltzmann equation with a rough force field (Arsénio and Saint-Raymond, in preparation [4]).

Original languageEnglish (US)
Pages (from-to)3044-3098
Number of pages55
JournalJournal of Functional Analysis
Volume261
Issue number10
DOIs
StatePublished - Nov 15 2011

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Keywords

  • Averaging lemma
  • Hypoellipticity
  • Kinetic theory
  • Microlocal decomposition
  • Transport equation

ASJC Scopus subject areas

  • Analysis

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