1. Rigorous study of the equilibria of collision kernels appearing in the theory of weak turbulence
- Author
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Maxime Breden, Laurent Desvillettes, Technische Universität Munchen - Université Technique de Munich [Munich, Allemagne] (TUM), Université Paris Diderot, Sorbonne Paris Cité, Paris, France, Université Paris Diderot - Paris 7 (UPD7), and The research leading to this paper was partly funded by Université SorbonneParis Cité, in the framework of the 'Investissements d'Avenir', convention ANR-11-IDEX-0005. MB also acknowledges partial support from a Lichtenberg Professorship grant ofthe VolkswagenStiftung awarded to C. Kuehn.
- Subjects
Keys words: weak turbulence ,Complex system ,82C40 ,FOS: Physical sciences ,Type (model theory) ,[MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA] ,01 natural sciences ,symbols.namesake ,equilibria Mathematical Subject Classification: 76F99 ,Mathematics - Analysis of PDEs ,Mathematics (miscellaneous) ,[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] ,FOS: Mathematics ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,Statistical physics ,0101 mathematics ,Mathematical Physics ,Physics ,76P05 ,Turbulence ,Mechanical Engineering ,010102 general mathematics ,Mathematical Subject Classification: 76F99, 76P05, 82C40 ,Turbulence theory ,Mathematical Physics (math-ph) ,Collision ,equilibria ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,010101 applied mathematics ,collision kernels ,Boltzmann constant ,Boltzmann's H-theorem ,symbols ,Particle ,weak turbulence ,Analysis ,Analysis of PDEs (math.AP) - Abstract
In this paper, we rigorously obtain all the equilibria of collision kernels of type ``two particles give two particles'' appearing in weak turbulence theory under very general assumptions, thus completing the ``equality case'' in Boltzmann's H-theorem for those models. We also provide some rigorous results for collision kernels of type ``two particles give one particle'', under assumptions which include some of the most classical kernels of this type. The method of proof is inspired by the quantitative estimates obtained for the Landau equation by Desvillettes in [J. Funct. Anal. 269:1359-1403, 2015].
- Published
- 2018