Regularity of optimal transport and cut locus: From nonsmooth analysis to geometry to smooth analysis
Cédric Villani
Discrete & Continuous Dynamical Systems - A 2011, 30(2): 559-571 doi: 10.3934/dcds.2011.30.559
In this survey paper I describe the convoluted links between the regularity theory of optimal transport and the geometry of cut locus.
keywords: fully nonlinear partial differential equations Ma-Trudinger-Wang curvature Optimal transport cut locus.
Entropy and chaos in the Kac model
Eric A. Carlen Maria C. Carvalho Jonathan Le Roux Michael Loss Cédric Villani
Kinetic & Related Models 2010, 3(1): 85-122 doi: 10.3934/krm.2010.3.85
We investigate the behavior in $N$ of the $N$--particle entropy functional for Kac's stochastic model of Boltzmann dynamics, and its relation to the entropy function for solutions of Kac's one dimensional nonlinear model Boltzmann equation. We prove results that bring together the notion of propagation of chaos, which Kac introduced in the context of this model, with the problem of estimating the rate of equilibration in the model in entropic terms, showing that the entropic rate of convergence can be arbitrarily slow. Results proved here show that one can in fact use entropy production bounds in Kac's stochastic model to obtain entropic convergence bounds for his non linear model Boltzmann equation, though the problem of obtaining optimal lower bounds of this sort for the original Kac model remains open and the upper bounds obtained here show that this problem is somewhat subtle.
keywords: Entropy propagation of chaos.
Celebrating Cercignani's conjecture for the Boltzmann equation
Laurent Desvillettes Clément Mouhot Cédric Villani
Kinetic & Related Models 2011, 4(1): 277-294 doi: 10.3934/krm.2011.4.277
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: relaxation to equilibrium entropy production Landau equation spectral gap Poincaré inequality. logarithmic Sobolev inequality relative entropy Cercignani's conjecture Boltzmann equation

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