VILLANI, CEDRIC / Mathematics / Researchers

International Center for Scientific Research

Researchers

Mathematics / VILLANI, CEDRIC


Position

Enseignant-chercheur en mathématiques
Unité de Mathématiques Pures et Appliquées (UMPA), Ecole Normale Supérieure de Lyon, Lyon, France.

Research interests

His primary research activity deals with partial differential equations of nonequilibrium statistical mechanics, in particular Boltzmann-like equations, and the Landau equation arising in plasma physics. He has been especially interested in the qualitative study of the solutions to these equations, including regularity theory and study of rates of convergence to equilibrium. Among my results in this field:

- the solution of the \'\'Cercignani conjecture\'\', a functional inequality relating the entropy and the entropy production for the Boltzmann equation (partially joint with Giuseppe Toscani);

- the solution of a conjecture by P.-L.~Lions, about the regularizing effects of grazing collisions in the Boltzmann equation, and the extension of the DiPerna-Lions theory to singular collision kernels (joint with Radjesvarane Alexandre);

- the first explicit estimates of convergence to equilibrium for very smooth solutions of the Boltzmann equatin, witout any assumption of smallness or linearization.


Then he led researches at a crosspoint between probability, functional analysis and partial differential equations: logarithmic Sobolev inequalities, functional concentration of measure, optimal transport, Wasserstein distances, information theory. Among his main results in this field :

- he showed close links between some concentration inequalities due to Talagrand, logarithmic Sobolev inequalities and diffusion equations (joint with Felix Otto); these works have been taken over and extended by many people.

- a new proof of optimal Sobolev inequalities and certain Gagliardo-Nirenberg interpolation inequalities, based on optimal transport (joint with Dario Cordero-Erausquin and Bruno Nazaret). He developed further these works with Francesco Maggi, which led them to the solution of an old open problem of Brézis and Lieb about optimal trace Sobolev inequalities.

- a synthetic definition of Ricci curvature lower bounds in metric-measure length spaces, the proof of stability for this definition, and its use to generalize various results from Riemannian geometry (joint with John Lott; closely related results were obtained independently by Karl-Theodor Sturm).


In collaboration with Emanuele Caglioti, José Carrillo, Irene Gamba, Robert McCann et Vladislav Panferov, he obtained various results about qualitative properties of kinetic equations of granular media; for instance the existence of equilibrium distributions with anomalously thick tails for a model of inelastic diffusive hard spheres, as conjectured by the physicist Matthieu Ernst.

His main collaborators: Radjesvarane Alexandre, Luigi Ambrosio, Alexander Bobylev, François Bolley, Yann Brenier, Eric Carlen, José Antonio Carrillo, Dario Cordero-Erausquin, Laurent Desvillettes Irene Gamba, Wilfrid Gangbo, François Golse, Alice Guionnet, Michel Ledoux and Pierre-Louis Lions, Michael Loss, John Lott, Francesco Maggi, Clément Mouhot, Felix Otto, Robert McCann, Giuseppe Toscani.

Prizes and awards

Jacques Herbrand Prize of the Academy of Sciences (2007)
Institut Universitaire de France (2006)
Invited lecturer at the International Congress of Mathematicians (2006)
Harold Grad lecturer (2004)
Plenary lecturer at the International Congress of Mathematical Physics (2003)
Peccot-Vimont Prize and Cours Peccot of the Collège de France (2003)
Louis Armand Prize of the Academy of Sciences (2001)
PhD Thesis (1998; advisor P.-L. Lions); Habilitation dissertation (2000)
Agrégation with rank 6 (1994).

Publications

Research papers

In collaboration with P.L. Lions: Régularité optimale de racines carrées. C.R. Acad. Sci. 321 (1995), 1537-1541.

On the Landau equation: weak stability, global existence. Adv. Diff. Eq. 1, 5 (1996), 793-816.

On the spatially homogeneous Landau equation for Maxwellian molecules. Math. Meth. Mod. Appl. Sci. 8, 6 (1998), 957-983.

Fisher information estimates for Boltzmann\'s collision operator. J. Maths Pures Appl. 77 (1998), 821-837.

On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations. Arch. Rat. Mech. Anal. 143, 3 (1998), 273-307.

Conservative forms of Boltzmann\'s collision operator: Landau revisited. Math. Mod. An. Num. 33, 1 (1999), 209-227.

In collaboration with G. Toscani: Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas. J. Statist. Phys. 94, 3/4 (1999), 619-637.

In collaboration with G. Toscani: Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation. Comm. Math. Phys. 203, 3 (1999), 667-706.

Regularity estimates via the entropy dissipation for the spatially homogeneous Boltzmann equation without cut-off. Rev. Matem. Iberoam. 15, 2 (1999), 335-352.

In collaboration with L. Desvillettes: On the spatially homogeneous Landau equation for hard potentials. Part I: Existence, uniqueness and smoothness. Comm. P.D.E 25, 1-2 (2000), 179-259.

In collaboration with L. Desvillettes: On the spatially homogeneous Landau equation for hard potentials. Part II: H-Theorem and applications. Comm. P.D.E 25, 1-2 (2000), 261-298.

Decrease of the Fisher information for solutions of the spatially homogeneous Landau equation with Maxwellian molecules. Math. Mod. Meth. Appl. Sci. 10, 2 (2000), 153-161.

In collaboration with G. Toscani: On the trend to equilibrium for some dissipative systems with slowly increasing a priori bounds. J. Statist. Phys. 98, 5-6 (2000), 1279-1309.

In collaboration with F. Otto: Generalization of an inequality by Talagrand, viewed as a consequence of the logarithmic Sobolev inequality. J. Funct. Anal. 173, 2 (2000), 361-400.

In collaboration with R. Alexandre, L. Desvillettes and B. Wennberg: Entropy dissipation and long-range interactions. Arch. Rat. Mech. Anal. 152 (2000), 327-355.

A short proof of the ``concavity of entropy power\'\'. IEEE Trans. Info. Theory 46, 4 (2000), 1695-1696.

In collaboration with L. Desvillettes: On the trend to global equilibrium in spatially inhomogeneous systems. Part I: the linear Fokker-Planck equation. Comm. Pure Appl. Math. 54, 1 (2001), 1-42.

In collaboration with F. Otto: Comment on : ``Hypercontractivity of Hamilton-Jacobi equations\'\', by S. Bobkov, I. Gentil and M. Ledoux. J. Math. Pures Appl. (9) 80, 7 (2001), 697-700.

In collaboration with R. Alexandre: On the Boltzmann equation for long-range interactions. Comm. Pure Appl. Math. 55, 1 (2002), 30-70.

In collaboration with E. Caglioti: Homogeneous cooling states are not always good approximations to granular flows. Arch. Rational Mech. Anal. 163, 4 (2002), 329-343.

In collaboration with L. Desvillettes: On a variant of Korn\'s inequality arising in statistical mechanics. A tribute to J.-L. Lions. ESAIM Control Optim. Calc. Var. 8 (2002), 603-619 (electronic).

In collaboration with L. Pareschi and G. Toscani: Spectral methods for the non cut-off Boltzmann equation and numerical grazing collision limit. Numer. Mat. 93, 3 (2003), 527-248.

Cercignani\'s conjecture is sometimes true and always almost true. Commun. Math. Phys. 234 (2003), 455-490 The original publication is available on LINK at http://link.springer.de or via the following link:

In collaboration with J.A. Carrillo and R. McCann: Kinetic equilibration rates for granular media and related equations: Entropy dissipation and mass transportation estimates. Rev. Matematica Iberoamericana 19 (2003), 1-48.

In collaboration with D. Cordero-Erausquin and B. Nazaret: A new approach to sharp Sobolev and Gagliardo-Nirenberg inequalities. Adv. Math. 182, 2 (2004), 307-332.

In collaboration with R. Alexandre: On the Landau approximation in plasma physics. Ann. Inst. H. Poincaré Anal. Non Linéaire 21, 1 (2004), 61-95.

In collaboration with I. Gamba et V. Panferov: On the Boltzmann equation for diffusively excited granular media. Comm. Math. Phys. 246, 3 (2004), 503-541.

In collaboration with C. Mouhot: Regularity theory for the spatially homogeneous Boltzmann equation with cut-off. Arch. Rational Mech. Anal. 173, 2 (2004), 35-43.

In collaboration with L. Desvillettes: On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation. Invent. Math. 159, 2 (2005), 245-316.

In collaboration with F. Maggi: Balls have the worst best Sobolev inequalities. To appear in J. Geom. Anal. 15, 1 (2005), 331-352

In collaboration with F. Bolley: Weighted Csiszár-Kullback-Pinsker inequalities and applications to transportation inequalities. Ann. Fac. Sci. Toulouse Math. 14, 3 (2005), 331-352.

In collaboration with J. Carrillo and R. McCann: Contractions in the 2-Wasserstein length space and thermalization of granular media. Arch. Rational Mech. Anal. 179 (2006), 217-263.

In collaboration with A. Guillin and F. Bolley: Quantitative concentration inequalities for empirical measures on non-compact spaces. Probab. Theory and Related Fields 137, 3-4 (2007), 287-314.

In collaboration with J. Lott: Weak curvature conditions and Poincaré inequalities. J. Funct. Anal. 245, 1 (2007), 311-333.

In collaboration with J. Lott: Hamilton-Jacobi semigroup on length spaces and applications. J. Math. Pures Appl. 88, 3 (2007), 219-229.

In collaboration with F. Maggi: Balls have the worst best Sobolev inequalities. Part II: Variants and extensions. Calc. Var. Partial Differential Equations 31, 1 (2008), 47-74.

In collaboration with A. Figalli: Strong displacement convexity on Riemannian manifolds. Math. Z. 257, 2 (2007), 251-259.

In collaboration with J. Lott: Ricci curvature for metric-measure spaces via optimal transport. Ann. of Math. (to appear).

Hypocoercivity. Memoirs Amer. Math. Soc. (to appear). (This memoir includes a debugged version of my obsolete text, \"Hypocoercive diffusion operators in Hörmander form\".)

In collaboration with N. Grunewald, F. Otto and M. Reznikoff: A two-scale approach to logarithmic Sobolev inequalities and the hydrodynamic limit. Ann. Inst. H. Poincaré Probab. Statist. (to appear).


Books

A review of mathematical topics in collisional kinetic theory. In Handbook of Mathematical Fluid Dynamics, S. Friedlander and D. Serre, Eds, Elsevier Science, 2002.

Limites hydrodynamiques de l\'équation de Boltzmann (d\'après C. Bardos, F. Golse, C. D. Levermore, P.-L. Lions, N. Masmoudi, L. Saint-Raymond). Bourbaki Seminar, Exp. 893 (June 2001). Astérisque 282 (2002), 365-405.

Topics in Optimal Transportation. Graduate Studies in Mathematics 58, American Mathematical Society, Providence (2003)

Optimal transportation, dissipative PDE\'s and functional inequalities. Notes for the CIME summer school ``Optimal transportation and applications\'\' (Martina Franca, September 2002). Lecture notes in mathematics, vol. 1813, L. Caffarelli and S. Salsa, Ed. © Springer-Verlag

Convergence to equilibrium: entropy production and hypocoercivity. Text of my Harold Grad lecture at the Rarefied Gas Dynamics 24 (Bari, June 2004).

Entropy production and convergence to equilibrium. Notes for a series of lectures in Institut Henri Poincaré, Paris (Winter 2001). Intended for publication in Lecture Notes in Mathematics.

Mathematics of granular materials. To appear in J. Stat. Phys.

Hypocoercive diffusion operators. Proceedings of the International Congress of Mathematicians (Madrid, 2006).

Optimal transport, old and new. To appear in Grundlehren der mathematischen Wissenschaften.

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