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Infinite energy solutions to the homogeneous Boltzmann equation
Author(s) -
Can Marco,
Karch Grzegorz
Publication year - 2010
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.20298
Subject(s) - mathematics , boltzmann equation , homogeneous , measure (data warehouse) , space (punctuation) , mathematical analysis , moment (physics) , kernel (algebra) , fourier transform , a priori and a posteriori , work (physics) , initial value problem , energy (signal processing) , a priori estimate , classical mechanics , physics , pure mathematics , quantum mechanics , combinatorics , linguistics , philosophy , statistics , epistemology , database , computer science
The goal of this work is to present an approach to the homogeneous Boltzmann equation for Maxwellian molecules with a physical collision kernel that allows us to construct unique solutions to the initial value problem in a space of probability measures defined via the Fourier transform. In that space, the second moment of a measure is not assumed to be finite, so infinite energy solutions are not a priori excluded from our considerations. Moreover, we study the large‐time asymptotics of solutions and, in a particular case, we give an elementary proof of the asymptotic stability of self‐similar solutions obtained by A. V. Bobylev and C. Cercignani [J. Stat. Phys. 106 (2002), 1039–1071]. © 2009 Wiley Periodicals, Inc.