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Celebrating Cercignani's conjecture for the Boltzmann equation

Laurent Desvillettes1, Clément Mouhot2, 3, Cédric Villani4

Cercignani's conjecture assumes a linear inequality between the entropy and entropy production functionals for Boltzmann's nonlinear integral operator in rarefied gas dynamics. Related to the field of logarithmic Sobolev inequalities and spectral gap inequalities, this issue has been at the core of the renewal of the mathematical theory of convergence to thermodynamical equilibrium for rarefied gases over the past decade. In this review paper, we survey the various positive and negative results which were obtained since the conjecture was proposed in the 1980s.

1:	CMLA - Centre de Mathématiques et de Leurs Applications
2:	DMA - Département de Mathématiques et Applications
3:	DPMMS/CMS
4:	ICJ - Institut Camille Jordan

Keyword(s) : Cercignani's conjecture – spectral gap – Boltzmann equation – relative entropy – entropy production – relaxation to equilibrium – Landau equation – logarithmic Sobolev inequality – Poincaré inequality

hal-00519608, version 1

http://hal.archives-ouvertes.fr/hal-00519608

oai:hal.archives-ouvertes.fr:hal-00519608

From: Clément Mouhot <>

Submitted on: Monday, 20 September 2010 23:41:38

Updated on: Tuesday, 21 September 2010 11:43:05