Kinetic and Related Models (KRM)

Celebrating Cercignani's conjecture for the Boltzmann equation

Pages: 277 - 294, Volume 4, Issue 1, March 2011      doi:10.3934/krm.2011.4.277

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Laurent Desvillettes - ENS Cachan, CMLA, IUF & CNRS, PRES UniverSud, 61, Av. du Pdt Wilson, 94235 Cachan Cedex, France (email)
Clément Mouhot - University of Cambridge, DPMMS, Wilberforce road, CB3 0WA, United Kingdom (email)
Cédric Villani - Institut Henri Poincaré & Université Claude Bernard Lyon 1, 11 rue Pierre et Marie Curie 75230 Paris Cedex 05, France (email)

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.

Keywords:  Cercignani's conjecture, spectral gap, Boltzmann equation, relative entropy, entropy production, relaxation to equilibrium, Landau equation, logarithmic Sobolev inequality, Poincaré inequality.
Mathematics Subject Classification:  26D10, 35A23, 76P05, 82C40, 82D10.

Received: September 2010;      Revised: October 2010;      Available Online: January 2011.