Celebrating Cercignani's conjecture for the Boltzmann equation

Desvillettes, Laurent; Mouhot, Clément; Villani, Cédric

Desvillettes, Laurent; Mouhot, Clément; Villani, Cédric (2011), Celebrating Cercignani's conjecture for the Boltzmann equation, Kinetic and related models, 4, 1, p. 277-294. http://dx.doi.org/10.3934/krm.2011.4.277

Type

Article accepté pour publication ou publié

External document link

http://arxiv.org/abs/1009.4006v2

Date

2011

Journal name

Kinetic and related models

Volume

Number

Publisher

American Institute of Mathematical Sciences

Pages

277-294

Abstract (EN)

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.

Subjects / Keywords

Cercignani's conjecture; spectral gap; Boltzmann equation; relative entropy; entropy production; relaxation to equilibrium; Landau equation; logarithmic Sobolev inequality; Poincaré inequality