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Regularity of renormalized solutions in the Boltzmann equation with long‐range interactions
Author(s) -
Arsénio Diogo,
Masmoudi Nader
Publication year - 2012
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21385
Subject(s) - mathematics , boltzmann equation , smoothness , kernel (algebra) , boltzmann constant , mathematical analysis , limit (mathematics) , range (aeronautics) , cutoff , a priori and a posteriori , dissipation , mathematical physics , combinatorics , physics , quantum mechanics , materials science , composite material , philosophy , epistemology
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 ${\textstyle{F \over {1 + F}}} \in W_{t,x,v}^{s,p}$ for every $1 \le p \le {\textstyle{D \over {D - 1}}}$ 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. © 2012 Wiley Periodicals, Inc.