Celebrating Cercignani's conjecture for the Boltzmann equation
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
TypeArticle accepté pour publication ou publié
External document linkhttp://arxiv.org/abs/1009.4006v2
Journal nameKinetic and related models
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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 / KeywordsCercignani's conjecture; spectral gap; Boltzmann equation; relative entropy; entropy production; relaxation to equilibrium; Landau equation; logarithmic Sobolev inequality; Poincaré inequality
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Large time behavior of the a priori bounds for the solutions to the spatially homogeneous Boltzmann equations with soft potentials. Desvillettes, Laurent; Mouhot, Clément (2007) Article accepté pour publication ou publié