Celebrating Cercignani's conjecture for the Boltzmann equation

Laurent Desvillettes; Clément Mouhot; Cédric Villani

doi:10.3934/krm.2011.4.277

Article Contents

2011, Volume 4, Issue 1: 277-294. Doi: 10.3934/krm.2011.4.277 open access

This issue Previous Article Kinetic approach to deflagration processes in a recombination reaction Next Article Computational high frequency wave diffraction by a corner via the Liouville equation and geometric theory of diffraction

Celebrating Cercignani's conjecture for the Boltzmann equation

1.
ENS Cachan, CMLA, IUF & CNRS, PRES UniverSud, 61, Av. du Pdt Wilson, 94235 Cachan Cedex
2.
University of Cambridge, DPMMS, Wilberforce road, CB3 0WA
3.
Institut Henri Poincaré & Université Claude Bernard Lyon 1, 11 rue Pierre et Marie Curie 75230 Paris Cedex 05

Received: September 2010

Revised: October 2010

Published: March 2011

Abstract / Introduction Related Papers Cited by

Abstract

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.

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Mathematics Subject Classification: 26D10, 35A23, 76P05, 82C40, 82D10.

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