Regularity of renormalized solutions in the Boltzmann equation with long-range interactions

Diogo Arsénio, Nader Masmoudi

Research output: Contribution to journalArticle

Abstract

It is well-established that renormalized solutions of the Boltzmann equation enjoy some kind of regularity, or at least compactness, in the velocity variable when the angular collision kernel is nonintegrable. However, obtaining explicit estimates in convenient and natural functional settings proves rather difficult. In this work, we derive a velocity smoothness estimate from the a priori control of the renormalized dissipation. As a direct consequence of our result, we show that, in the presence of long-range interactions, any renormalized solution F(t, x, v) to the Boltzmann equation satisfies locally F/1 + F ∈ W- t,x,v s,p for every 1 le; p < D/D - 1 and for some s > 0 depending on p. We also provide an application of this new estimate to the hydrodynamic limit of the Boltzmann equation without cutoff.

Original languageEnglish (US)
Pages (from-to)508-548
Number of pages41
JournalCommunications on Pure and Applied Mathematics
Volume65
Issue number4
DOIs
StatePublished - Apr 2012

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Renormalized Solutions
Boltzmann equation
Long-range Interactions
Boltzmann Equation
Regularity
Estimate
Hydrodynamic Limit
Compactness
Dissipation
Smoothness
Hydrodynamics
Collision
kernel

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Regularity of renormalized solutions in the Boltzmann equation with long-range interactions. / Arsénio, Diogo; Masmoudi, Nader.

In: Communications on Pure and Applied Mathematics, Vol. 65, No. 4, 04.2012, p. 508-548.

Research output: Contribution to journalArticle

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