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

Research output: Contribution to journalArticlepeer-review


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
Issue number4
StatePublished - Apr 2012

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


Dive into the research topics of 'Regularity of renormalized solutions in the Boltzmann equation with long-range interactions'. Together they form a unique fingerprint.

Cite this