## Surveys / Books

Below is a selection of research-level texts which I have written for pedagogical purposes. They are mainly focused on recent research, and describe other researchers’ contributions as well as mine. Here I only put those texts which I consider most significant; the other ones can be found in my publication list.

### A Review of Mathematical Topics in Collisional Kinetic Theory

This is a book-size review paper about kinetic models describing collisions between particles, such as Boltzmann’s equation and its variants. The emphasis is on the mathematical point of view, and many recent contributions are reviewed as well as old problems. Chapter 1 describes models and problems, Chapter 2 is about the theory of the Cauchy problem (including regularity and estimates on solutions), Chapter 3 enters into the links and analogies between kinetic theory and information theory, Chapter 4 is concerned with the trend to thermodynamical equilibrium. Finally, Chapter 5 reviews classical open problems as well as some new trends: the theory of granular media and the development of mathematical models in quantum kinetic theory.

This text was mostly written in the end of year 2000. It appeared in 2002, in Volume I of the *Handbook of mathematical fluid mechanics* edited by Susan Friedlander and Denis Serre, published by North-Holland. This volume gathers several expository papers about compressible gas dynamics, including a more physical introduction to the Boltzmann equation, written by Carlo Cercignani.

Around 2005 I felt the need to rewrite a synthesis text on granular media; indeed, the treatment of this subject in my survey paper had become a bit outdated, because of the rapid expansion of the field.

You can find here the text in compressed form (which is natural since we are dealing with compressible fluid dynamics): PDF File

A few mistakes have been reported to me and are reviewed in the errata page.

### Topics in Optimal Transportation

This book is an introduction to the Monge-Kantorovich minimization problem and its ramifications in various branches of mathematics. Intended to serve both as a survey of recent research and as a textbook for students, it covers both the theory (Kantorovich duality, characterization of optimizers, Monge-Ampère equation, Monge-Kantorovich-Wasserstein distances), the applications to fluid mechanics (like Brenier’s polar factorization theorem and its consequences), geometrico-functional inequalities (Brunn-Minkowski, Sobolev, etc.), and the study of dissipative equations (long-time behavior of some gradient flow systems). Many exercises and problems are included.

It took me three years to write this book, which initially was intended to be just lecture notes for students of a graduate course. It is published by the American Mathematical Society in the Graduate Studies in Mathematics series, vol. 58 (2003).

A short presentation can be found on the Web site of the AMS Bookstore. The Table of Contents and the Preface can be downloaded here: PDF File

You can also consult the list of errata.

### Limites hydrodynamiques de l’équation de Boltzmann (Séminaire Bourbaki)

This is the text of the Bourbaki seminar which I gave in June 2001 to present the works of Claude Bardos, François Golse, Dave Levermore, Pierre-Louis Lions, Nader Masmoudi and Laure Saint-Raymond about the incompressible hydrodynamical limit of the Boltzmann equation. The joint efforts of these people led to a most impressive mathematical achievement in fluid mechanics: a rigorous and general proof that solutions of the Boltzmann equation do reduce, in a certain physical regime, to solutions of the incompressible Navier-Stokes equations. Even though the results can be hoped to be improved in many respects (these are only statements about rescaled sequences of solutions, the proof is not constructive and does not cover physically realistic interactions), these contributions are an important step towards the accomplishment of one of the dreams of Maxwell, Boltzmann and Hilbert: the derivation of the basic equations of fluid mechanics from atomistic models. Besides, they construct a bridge between two of the most famous theories of weak solutions in fluid mechanics: the DiPerna-Lions and Leray theories.

Written in french, this text tries to explain the main ideas and results to a non-expert reader. After an elementary introduction to the problem and its motivations, I describe the Bardos-Golse-Levermore program as it was formalized ten years ago; then I sketch several important estimates including entropy estimates, the analysis of acoustic waves, and weak compactness methods via the “regularity of the gain operator” and “averaging lemmas”.

Like all Bourbaki seminars, this text appeared in the french journal *Astérisque*. It can be downloaded here: PDF File

### Optimal transportation, dissipative PDE’s and functional inequalities (CIME Lecture Notes)

This is a set of lecture notes written for the CIME summer school “Optimal transportation and applications” (Martina Franca, 2000). There I discuss various problems which have been solved recently by means of optimal transportation. After reviewing some motivations which apparently have nothing to do with optimal transportation, I present an example of study of fast trend to equilibrium for a nonlinear, nonlocal partial differential equation arising in the modelling of granular media; then another example in which it is proven that convergence is necessarily slow; then I illustrate how optimal transportation and concentration estimates can be used in a problem of mean-field limit, following the recent PhD thesis of Malrieu; finally I explain a bit about Otto’s differential point of view on optimal transportation and its applications. The style is rather informal and the presentation departs from that of a standard course in the sense that the theory comes at the same time as, or even after, the applications. Most of the material in sections III and IV is not present in the main text of my book *Topics in Optimal Transportation*, but has been added in the form of problems at the end.

These notes appeared in the *Lecture notes in mathematics* series by Springer. The text can be downloaded here: PDF File **© Springer-Verlag**

### Entropy production and convergence to equilibrium (Lecture Notes from a Course at IHP)

This is a set of lecture notes written on the occasion of a course at Institut Henri Poincaré in the Winter of 2001; the text was completed and updated in the beginning of 2004, then slighly updated again for its publication in 2007.

These notes develop some of the topics addressed in my Review of Mathematical Topics in Collisional Kinetic Theory, the main topic being the use of fine entropy production techniques to study the convergence to equilibrium for the Boltzmann equation by constructive methods. I included some reminders about information theory, and about the theory of the Cauchy problem for the Boltzmann equation. Some current directions of research, in particular about linearization, spectral gap and exponential convergence revisited, are discussed in the end of the notes.

This course was part of a thematic semester about Hydrodynamic Limits and related topics, organized by François Golse and Stefano Olla. It appeared in the Lecture notes in mathematics series by Springer, together with the course of Fraydoun Rezakhanlou in the same program.

The text can be downloaded here: PDF File

### Convergence to equilibrium: Entropy production and hypocoercivity

This is the text for my Harold Grad lecture, delivered in Bari on the occasion of the 24th RGD meeting, in June 2004 (see the beautiful cake, photographed by Kazuo Aoki!). The themes are pretty much the same as the ones in the Henri Poincaré lectures, but this is a shorter and more lively text, focusing both on historical issues and very recent research results.

It can be downloaded here: PDF File

### Mathematics of granular material

This is a short and slightly informal survey on the kinetic theory of granular material, published as part of a special issue of Journal of Statistical Physics dedicated to Carlo Cercignani. (It was a great honor for me to write the preface of this volume.) The kinetic modelling of granular material was already considered in my survey paper on collisional kinetic theory; but since then the theory had grown so fast that I felt the need for another synthesis text of the subject.

The text can be downloaded here: PDF File

### Hypocoercive diffusion operators

This is my text for the proceedings of the 2006 International Congress of Mathematicians in Madrid. It reviews some recent trends in the study of convergence to equilibrium for various degenerate equations, either linear or nonlinear, which are reminiscent of the classical theory of hypoellipticity. Several motivations from various domains of physics are exposed.

The text can be downloaded here: PDF File

### Optimal Transport, Old and New

This book is an expanded version of my lecture notes from Saint-Flour 2005. This project, which took up more or less two years of my life :-), can be considered as the sequel of my first book on optimal transport (TOT = Topics in Optimal Transportation), although it can be read independently, and relies on quite different choices of presentation.

With respect to **TOT**, the present book is longer, more detailed and more advanced. From the beginning it deals with much quite general metric structures (metric spaces and Riemannian manifolds rather than just Euclidean space). The presentation is more focused on geometry, probability and dynamical systems. Many Appendices have been added to make the text as self-contained as possible. Complete proofs of the most important results are available there, with the best sets of assumptions known at the time of writing. A number of statements and theorems have been proven specifically for this course. There are tentative extensive bibliographical notes.

This text is intended to be used both by beginners who look for a self-contained introduction, and by more advanced readers as a reference text. It has appeared (in a beautiful edition) in Springer’s Grundlehren der mathematischen Wissenschaften collection. It is downloadable here: PDF File

### Optimal transport and curvature

This text, written in collaboration with Alessio Figalli, records the lectures which I gave in the CIME Summer Course in Cetraro, June 2008. The general theme is the interrelation between optimal transport and Riemann curvature. The geometric part is mainly self-contained, and slightly computation-oriented. After some reminders about Riemannian geometry and the basic core of optimal transport theory, there are two independent parts devoted respectively to lower Ricci curvature bounds, and Ma-Trudinger-Wang curvature bounds. A final part records some recent trends and open problems.

The text can be downloaded here: PDF File

### Paradoxe de Scheffer-Shnirelman revu sous l’angle de l’intégration convexe (Séminaire Bourbaki)

This is the text of the Bourbaki seminar which I gave in November 2008 to present the works of Camillo De Lellis and Laszló Székelyhidi, who dramatically clarified and improved the famous Scheffer-Shnirelman paradox according to which there are nontrivial weak solutions of the incompressible Euler equation which have compact support in time.

This lecture, written in French, recalls the history of the subject, and explains in detail the connection which the authors found with Gromov’s convex integration and Tartar’s wave analysis. Then I present complete proofs of the main results, and conclude with the presentation of some developments and speculations. The text can be downloaded here: PDF File

### Landau damping

In this section I have gathered two texts summarizing my work with Clément Mouhot on Landau damping in the nonlinear regime (or equivalently, for very large times), according to which Landau damping survives nonlinearity, by a phenomenon which is reminiscent of the famous KAM Theorem in dynamical systems.

The first text is a very short summary written for the 50th anniversary of the Journal of Mathematical Physics (JMP). Our work echoes the foundations of JMP, by answering a question raised by Backus in the first mathematically rigorous treatment of (linear) Landau damping, which appeared in the very first volume of JMP.

Download here: PDF File

The second text, much longer, is a set of the lecture notes for a course of 5-6 hours which I taught in Cotonou (Benin) and in CIRM (Luminy, France). It starts from the basic notions of mean field limits and provides substantial bibliographical notes.

Download here: PDF File

Both texts are intended to be read by a public of physicists and mathematicians. The lecture notes include improved results which were not available at the time when the JMP note was written, but the main ingredients are already in the JMP note.

### (Ir)réversibilité et entropie

The text below (in French) was written for the Poincaré (“Bourbaphy”) seminar from November 2010, devoted to Time. It surveys some of the basics of the notions of arrow of time, reversibility/irreversibility, and entropy. Download here the French version: PDF File — or the English version: PDF File