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Fluid dynamic limits of kinetic equations II convergence proofs for the boltzmann equation
Author(s) -
Bardos Claude,
Golse François,
Levermore C. David
Publication year - 1993
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.3160460503
Subject(s) - mathematics , infinitesimal , conservation law , boltzmann equation , limit (mathematics) , mathematical analysis , boltzmann constant , compact space , kinetic energy , convergence (economics) , kullback–leibler divergence , momentum (technical analysis) , sequence (biology) , weak convergence , limit of a sequence , classical mechanics , physics , thermodynamics , statistics , genetics , computer security , finance , biology , computer science , economics , asset (computer security) , economic growth
Abstract Using relative entropy estimates about an absolute Maxwellian, it is shown that any properly scaled sequence of DiPerna‐Lions renormalized solutions of some classical Boltzmann equations has fluctuations that converge to an infinitesimal Maxwellian with fluid variables that satisfy the incompressibility and Boussinesq relations. Moreover, if the initial fluctuations entropically converge to an infinitesimal Maxwellian then the limiting fluid variables satisfy a version of the Leray energy inequality. If the sequence satisfies a local momentum conservation assumption, the momentum densities globaly converge to a solution of the Stokes equation. A similar discrete time version of this result holds for the Navier‐Stokes limit with an additional mild weak compactness assumption. The continuous time Navier‐Stokes limit is also discussed. © 1993 John Wiley & Sons, Inc.