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Transport optimal et courbure de Ricci

Cédric Villani

Séminaire de théorie spectrale et géométrie

Des liens inattendus ont été récemment mis à jour entre le transport optimal de Monge–Kantorovich et certains problèmes de géométrie riemannienne, en liaison avec la courbure de Ricci. Une des retombées de ces interactions est la naissance d’une théorie « synthétique » des espaces métriques mesurés à courbure de Ricci minorée, venant compléter la théorie classique des espaces métriques à courbure sectionnelle minorée. Dans ce texte (également fourni aux actes du Séminaire d’Équations aux dérivées...

Transport optimal et courbure de Ricci

Cédric Villani

Séminaire Équations aux dérivées partielles

Des liens inattendus ont été récemment mis à jour entre le transport optimal de Monge–Kantorovich et certains problèmes de géométrie riemannienne, en liaison avec la courbure de Ricci. Une des retombées de ces interactions est la naissance d’une théorie “synthétique” des espaces métriques mesurés à courbure de Ricci minorée, venant compléter la théorie classique des espaces métriqes à courbure sectionnelle minorée. Dans ce texte (également fourni aux actes du Séminaire de Théorie Spectrale et Géométrie...

Regularity estimates via the entropy dissipation for the spatially homogeneous Boltzmann equation without cut-off.

Cédric Villani — 1999

Revista Matemática Iberoamericana

We show that in the setting of the spatially homogeneous Boltzmann equation without cut-off, the entropy dissipation associated to a function f ∈ L(R) yields a control of √f in Sobolev norms as soon as f is locally bounded below. Under this additional assumption of lower bound, our result is an improvement of a recent estimate given by P.-L. Lions, and is optimal in a certain sense.

Hypocoercive Diffusion Operators

Cédric Villani — 2007

Bollettino dell'Unione Matematica Italiana

In many problems coming from mathematical physics, the association of a degenerate diffusion operator with a conservative operator may lead to dissipation in all variables and convergence to equilibrium. One can draw an analogy with the well-studied phenomenon of hypoellipticity in regularity theory, and actually both phenomena have been studied together. Now a distinctive theory of ``hypocoercivity'' is starting to emerge, with already some striking results, and several challenging open problems....

On a variant of Korn’s inequality arising in statistical mechanics

L. DesvillettesCédric Villani — 2002

ESAIM: Control, Optimisation and Calculus of Variations

We state and prove a Korn-like inequality for a vector field in a bounded open set of N , satisfying a tangency boundary condition. This inequality, which is crucial in our study of the trend towards equilibrium for dilute gases, holds true if and only if the domain is not axisymmetric. We give quantitative, explicit estimates on how the departure from axisymmetry affects the constants; a Monge–Kantorovich minimization problem naturally arises in this process. Variants in the axisymmetric case are...

On a variant of Korn's inequality arising in statistical mechanics

L. DesvillettesCédric Villani — 2010

ESAIM: Control, Optimisation and Calculus of Variations

We state and prove a Korn-like inequality for a vector field in a bounded open set of N , satisfying a tangency boundary condition. This inequality, which is crucial in our study of the trend towards equilibrium for dilute gases, holds true if and only if the domain is not axisymmetric. We give quantitative, explicit estimates on how the departure from axisymmetry affects the constants; a Monge–Kantorovich minimization problem naturally arises in this process. Variants in the axisymmetric case...

A two-scale approach to logarithmic Sobolev inequalities and the hydrodynamic limit

Natalie GrunewaldFelix OttoCédric VillaniMaria G. Westdickenberg — 2009

Annales de l'I.H.P. Probabilités et statistiques

We consider the coarse-graining of a lattice system with continuous spin variable. In the first part, two abstract results are established: sufficient conditions for a logarithmic Sobolev inequality with constants independent of the dimension (Theorem 3) and sufficient conditions for convergence to the hydrodynamic limit (Theorem 8). In the second part, we use the abstract results to treat a specific example, namely the Kawasaki dynamics with Ginzburg–Landau-type potential.

Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates.

José A. CarrilloRobert J. McCannCédric Villani — 2003

Revista Matemática Iberoamericana

The long-time asymptotics of certain nonlinear , nonlocal, diffusive equations with a gradient flow structure are analyzed. In particular, a result of Benedetto, Caglioti, Carrillo and Pulvirenti [4] guaranteeing eventual relaxation to equilibrium velocities in a spatially homogencous model of granular flow is extended and quantified by computing explicit relaxation rates. Our arguments rely on establishing generalizations of logarithmic Sobolev inequalities and mass transportation inequalities,...

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