Cédric Villani

Chapter 2 – A Review of Mathematical Topics in Collisional Kinetic Theory

This text has appeared with minor modifications in the Handbook of Mathematical Fluid Dynamics (Vol. 1), edited by S. Friedlander and D. Serre, published by Elsevier Science (2002). I have performed some last-minute addenda and corrections to take into account very recent advances; they are put as footnotes within the body of the text. Some other corrections have been performed after publication and are incorporated within the text, without modification of the numbering of formulas, subsections or footnotes. These modifications are listed on the Errata sheet for this text, which can be downloaded from

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

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, via either the Bakry-Emery method or the abstract approach of Otto and Villani [28].

A MASS-TRANSPORTATION APPROACH TO SHARP SOBOLEV AND GAGLIARDO-NIRENBERG INEQUALITIES

We show that mass transportation methods provide an elementary and powerful approach to the study of certain functional inequalities with a geometric content, like sharp Sobolev or Gagliardo-Nirenberg inequalities. The Euclidean structure of Rn plays no role in our approach: we establish these inequalities, together with cases of equality, for an arbitrary norm.

On Landau damping

Going beyond the linearized study has been a longstanding problem in the theory of Landau damping. In this paper we establish exponential Landau damping in analytic regularity. The damping phenomenon is reinterpreted in terms of transfer of regularity between kinetic and spatial variables, rather than exchanges of energy; phase mixing is the driving mechanism. The analysis involves new families of analytic norms, measuring regularity by comparison with solutions of the free transport equation; new functional inequalities; a control of non-linear echoes; sharp “deflection” estimates; and a Newton approximation scheme. Our results hold for any potential no more singular than Coulomb or Newton interaction; the limit cases are included with specific technical effort. As a side result, the stability of homogeneous equilibria of the non-linear Vlasov equation is established under sharp assumptions. We point out the strong analogy with the KAM theory, and discuss physical implications. Finally, we extend these results to some Gevrey (non-analytic) distribution functions.

On the trend to global equilibrium in spatially inhomogeneous entropy‐dissipating systems: The linear Fokker‐Planck equation

We study the long-time behavior of kinetic equations in which transport and spatial confinement (in an exterior potential or in a box) are associated with a (degenerate) collision operator acting only in the velocity variable. We expose a general method, based on logarithmic Sobolev inequalities and the entropy, to overcome the well-known problem, due to the degeneracy in the position variable, of the existence of infinitely many local equilibria. This method requires that the solution be somewhat smooth. In this paper, we apply it to the linear Fokker-Planck equation and prove decay to equilibrium faster than O(t−1/ϵ) for all ϵ > 0.

On the spatially homogeneous landau equation for hard potentials part i : existence, uniqueness and smoothness

We study the Cauchy problem for the homogeneous Landau equation of kinetic theory, in the case of hard potentials. We prove that for a large class of initial data, there exists a unique weak solution to this problem, which becomes immediately smooth and rapidly decaying at infinity.

Probability Metrics and Uniqueness of the Solution to the Boltzmann Equation for a Maxwell Gas

We consider a metric for probability densities with finite variance on ℝd, and compare it with other metrics. We use it for several applications both in probability and in kinetic theory. The main application in kinetic theory is a uniqueness result for the solution of the spatially homogeneous Boltzmann equation for a gas of true Maxwell molecules.

Regularity theory for the spatially homogeneous Boltzmann equation with cut-off

Abstract.We develop the regularity theory of the spatially homogeneous Boltzmann equation with cut-off and hard potentials (for instance, hard spheres), by (i) revisiting the Lp theory to obtain constructive bounds, (ii) establishing propagation of smoothness and singularities, (iii) obtaining estimates on the decay of the singularities of the initial datum. Our proofs are based on a detailed study of the “regularity of the gain operator”. An application to the long-time behavior is presented.

On the Boltzmann Equation for Diffusively Excited Granular Media

We study the Boltzmann equation for a space-homogeneous gas of inelastic hard spheres, with a diffusive term representing a random background forcing. Under the assumption that the initial datum is a nonnegative L2(N) function, with bounded mass and kinetic energy (second moment), we prove the existence of a solution to this model, which instantaneously becomes smooth and rapidly decaying. Under a weak additional assumption of bounded third moment, the solution is shown to be unique. We also establish the existence (but not uniqueness) of a stationary solution. In addition we show that the high-velocity tails of both the stationary and time-dependent particle distribution functions are overpopulated with respect to the Maxwellian distribution, as conjectured by previous authors, and we prove pointwise lower estimates for the solutions.

On the spatially homogeneous landau equation for hard potentials part ii : h-theorem and applications

We find a lower bound for the entropy dissipation of the spatially homogeneous Landau equation with hard potentials in terms of the entropy itself. We deduce from this explicit estimates on the speed of convergence towards equilibrium for the solution of this equation. In the case of so-called overmaxwellian potentials, the convergence is exponential. We also compute a lower bound for the spectral gap of the associated linear operator in this setting.