Presentation of my Research
Below is a selection of some of my most significant research papers (the choice is of course subjective). All the rest of my papers can be found in my publication list. There you can also find electronic links to the journals in which these papers have been published.
I divided this selection (sometimes a bit artificially) into various themes and sub-themes:
Collisional kinetic theory (Boltzmann type)
- Regularizing effects of grazing collisions
- Grazing collisions limit
- Cercignani entropy production conjecture
- Space-inhomogeneous convergence to equilibrium
- Qualitative properties of kinetic models for granular media
Collisionless kinetic theory (Vlasov type)
- Optimal transport, Ricci curvature and geometric inequalities
- Optimal transport and isoperimetric/Sobolev inequalities
- Regularity of optimal transport in curved geometry
In the explanations, I shall recall only basic facts and I shall not give precise references. A lot more information can be found in the papers themselves, as well as in my surveys, books and lecture notes.
A remark about terminology: for a long time I followed a common practice in kinetic theory, compressible fluid mechanics and probability theory, which is to call “entropy” the negative of the physical entropy. After some time, and contact with other communities, I decided that this was ridiculous and switched to the other convention, so what I call entropy is really the physical entropy. This is the reason why my older papers study “entropy dissipation”, while the newer ones care about “entropy production”…
Regularizing effects of grazing collisions in kinetic equations
This is a theme on which I have been working since the beginning of my PhD. The modelling of long-range interactions with a Boltzmann-type operator leads to a nonintegrable singularity for very small deviation angles in Boltzmann’s collision kernel. This means that most collisions involve a very, very small deviation of colliding particles. This singularity occurs as soon as your interaction has infinite range, even if it decays very fast at infinity. To avoid the numerical or theoretical difficulties caused by this singularity, it is customary in physics and in modelling to truncate the collision kernel: this is called Grad’s angular cut-off, and as a consequence the original equation is usually called the Boltzmann equation without cut-off. Of course one may wonder whether this truncation has changed the properties of the Boltzmann equation. For some topics (e.g. Euler fluid approximation) this has apparently no influence. But for other properties things change drastically; in the first place, regularity properties of the solution are heavily affected by the truncation.
If you think of the evolution by Boltzmann’s equation in a stochastic way, the abundance of very small deviations should make the process look not like a pure jump process, but rather like something intermediate between pure diffusion and pure jump. If you think of it in an way, you can believe that the Boltzmann operator with the singularity will act like a singular (fractional differential) operator. Either way you like to think about it, this extra diffusivity property should result in additional smoothness properties of the solutions, which are destroyed by the cutoff procedure. This guess was formulated explicitly by P.-L. Lions at the beginning of the nineties. A few years later, Desvillettes managed to prove that certain simple spatially homogeneous models without cut-off do enjoy regularizing properties, in a very strong sense. Sylvie Méléard and collaborators developed alternative probabilistic methods to study similar effects. More details can be found in my review on collisional kinetic theory.
However, until the end of the nineties, there was no mathematical evidence of this regularizing effect of grazing collisions in the spatially inhomogeneous situation. In the most general case, the entropy production estimate was essentially the only known a priori estimate; therefore it was natural to search for a way to express the regularity induced by grazing collisions in terms of entropy production. Various preliminary results on this problem were obtained first by Alexandre, then by P.-L. Lions, then by myself. But the first neat and optimal estimates were established in the following work, which in a way was the starting point of the “modern” theory of grazing collisions:
R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg: Entropy dissipation and long-range interactions. Arch. Rat. Mech. Anal. 152 (2000), 327-355. PDF File
The main result establishes that the entropy production functional (there called entropy dissipation functional) controls a precise fractional Sobolev norm of the distribution function, as soon as one has a moderate control of the mass, kinetic energy and entropy. Of course this result deals only with distribution functions of a velocity variable, but this is in the order of things since the entropy production functional only acts on the velocity variable in Boltzmann’s equation. In this paper were identified several tools which proved to be very useful in subsequent studies of grazing collisions: in particular a precise version of a symmetry-based cancellation effect (there called “cancellation lemma”) by which one can control the singularity in some situations; and a tricky variant of Parseval’s formula which enabled us to derive simple and optimal estimates in Fourier space for some Dirichlet forms.
To deduce anything about the full Boltzmann equation without cut-off and in the large, there was need of another tool that would help control the nasty nonlinearity in Boltzmann’s equation. Under the cut-off assumption, such a tool was introduced by the work of Ron DiPerna and P.-L. Lions at the end of the eighties: it is known as renormalization, and is more or less a recipe for performing nonlinear changes of unknown in the equation. The DiPerna-Lions renormalization was however strongly based on the cut-off assumption, and a completely different recipe was needed. For ten years, this problem had resisted the imagination of all specialists in the field; credit for unlocking it must be attributed to Alexandre, who found such a formulation in 1999. Shortly after, I was able to simplify and extend the validity of his formulation, and together we proved the first general result of existence of renormalized solutions for the Boltzmann equation without cut-off. This is the content of the following paper:
R. Alexandre and C. Villani: On the Boltzmann equation for long-range interactions. Comm. Pure Appl. Math. 55, 1 (2002), 30-70. PDF File
There we prove an extremely general version of the regularizing effect, in a form that had been conjectured by Lions: take a sequence of solutions of the Boltzmann equation without cut-off, satisfying certain natural a priori bounds: bounded entropy, bounded energy, bounded mass. Then this sequence, a priori precompact in the weak topology, is in fact precompact in the strong topology.
I will not say more on the subject, but just mention that the field of grazing collisions has since then continued to grow with the efforts of Desvillettes, Alexandre, and a number of other collaborators. Some recent developments involve harmonic-analysis tools and pseudo-differential calculus.
Grazing collisions limit
The grazing collisions approximation is well-known in plasma physics and goes back to a seminal 1936 paper by Landau. The issue is about the replacement of Boltzmann’s collision operator by Landau’s collision operator, when “almost all” collisions are grazing. I obtained very general results on this problem in my PhD, for the spatially homogeneous Boltzmann equation, by various methods. But with the new developments in the field of grazing collisions, Alexandre and I had all the ingredients to solve this problem in the considerably more tricky case of spatially inhomogeneous gas.
For Coulomb interaction the problem was very challenging because of the strong singularities involved: besides the angular singularity associated with grazing collisions, there was the singularity in the kinetic dependence of the collision kernel – which even in the cutoff case was outside the scope of the DiPerna-Lions theory! It turned out, to our surprise, that our new renormalized formulation was able to handle both singularities at the same time (well… borderline for both!)
With these tools and some additional sweat, we established the validity of the approximation in the most general case:
R. Alexandre and C. Villani: On the Landau approximation in plasma physics. Ann. Inst. H. Poincaré Anal. Non Linéaire 21, 1 (2004), 61-95. PDF File
There is however an important caveat: this limit is not the physically relevant regime if there is a self-consistent field. Well, you can always add the self-consistent field and get a perfectly sound mathematical problem, which you can solve by our method, but this is not the physically relevant regime, as is described at length in the paper. And in fact there is no physically relevant limit leading from Boltzmann to Landau, at least for classical Coulomb plasmas: the problem should be formulated not as a limit, but as a higher-order correction to the limit. In particular, it can be expressed only in terms of estimates, and requires some kind of “classical” control on the solution. To summarize: although the results in our paper are the best that one can dream of in the framework of limits of weak solutions, they cannot solve the physically relevant problem.
From this story I learnt two important things:
- Going from one model to another is not just all about limits;
- Weak solutions sometimes are not enough to do physics.
Estimating the rates of convergence to equilibrium for certain evolution equations has been one of my favourite topics. Among other things, it is crucial to determine whether the assumption of thermodynamical equilibrium is justified or not. And anyhow, the convergence problem is a beautiful part of nonequilibrium statistical mechanics, likely to hide some mathematical and physical gems.
All my work on convergence to equilibrium has been concerned with finding constructive estimates, rather than just using soft compactness and functional analytic tools. This imposed constraint makes things all of a sudden more difficult (and more interesting) by several orders of magnitude.
Convergence to equilibrium for solutions of Boltzmann’s equation is usually explained by Boltzmann’s H theorem: the entropy functional cannot decrease with time, and in fact it can only increase unless the distribution function is locally at equilibrium, which means that at each position the velocity profile is of Maxwellian, or hydrodynamic type. If the gas is enclosed in a box (without axis of symmetry), then a solution of Boltzmann equation which is locally at equilibrium has to be globally at equilibrium, i.e. the parameters of the local Maxwellian should be homogeneous (uniform density, uniform temperature, zero macroscopic velocity). It also turns out that such a state of global equilibrium is a state of maximum entropy for the distribution function, given the total mass and kinetic energy which are kept constant by the time-evolution. So anybody would believe that solutions of Boltzmann’s equation do converge to such a state as time becomes large, and this would justify the maximum entropy principle for this model. This conclusion is indeed correct, as soon as the solution satisfies a few simple a priori estimates (say bounded in some weighted Lebesgue space, uniformly in time – by the way, so far such estimates have been established only in very particular situations). But this tells nothing about rates of convergence!
Apparently the first people to have been concerned with rates of convergence for the Boltzmann equation are Mark Kac and Henry McKean. In the sixties, the latter did obtain some explicit rates of convergence for a simple one-dimensional, spatially homogeneous caricature of the Boltzmann equation, called the Kac caricature. Building on an idea by Kac, McKean demonstrated that Fisher’s quantity of information could be used successfully in kinetic theory. However, it is only in the early nineties that more general results appeared, with two influential papers by Eric Carlen and Maria Carvalho. They established that Boltzmann’s entropy production functional controls the distance (in the sense of Kullback information, or relative entropy) of the distribution function to the (local) Maxwellian equilibrium. The control was quite weak and complicated, but these results were the very first of their kind. Much more on the history of the subject can be found in my lecture notes on entropy production.
Even with all the smoothness in the world, the study of convergence to equilibrium for the Boltzmann equation turns out to be very tricky; among other things because of the complexity of the Boltzmann collision operator, the degeneracy of the Boltzmann equation in the position variable (collisions only act on velocity distribution, locally at each position), and the existence of three local conservation laws (mass, momentum and energy). A first step in this program consisted in improving the entropy inequalities by Carlen and Carvalho. An efficient strategy for this improvement was devised by Giuseppe Toscani and myself around 1997: it rests on Ornstein-Uhlenbeck regularization, a careful (competing) symmetry study, a bit of information theory, and a lower bound for the Landau entropy production functional, which I had established with Laurent Desvillettes a few years before. Our discovery (completely unexpected, as far as I am concerned) was that, roughly speaking, “the dissipation of the Boltzmann entropy production by the Fokker-Planck semigroup resembles the Landau entropy production”.
Around 2003 I improved these results with more technical estimates, and proved that
- for collision kernels that are bounded below and grow quadratically in large relative velocities, one may bound below the entropy production functional by a multiple of the relative entropy (i.e. the difference between the entropy of the Maxwellian equilibrium and that of the distribution function), as soon as the distribution function is spread enough (not concentrated on a line). This was the first situation where the “Cercignani conjecture” could be established. For more general collision kernels, e.g. with subquadratic growth, the inequality is known to be false! (I have my own conjectures about the domain of validity)
- Still, for these more general collision kernels, the entropy production functional may be bounded below by a power of the relative entropy, with exponent arbitrarily close to 1, as soon as one has a good control of the smoothness of the distribution function.
All this, together with precise references, can be found in the following paper:
C. Villani: Cercignani’s conjecture is sometimes true and always almost true. Commun. Math. Phys. 234 (2003), 455-490. PDF File
In addition, I could establish some new entropy production estimates for a multi-particle variant of the Kac caricature (a “master equation”), of which Kac believed that it could bring some insights into the trend to equilibrium. When I wrote this paper, the spectral gap problem associated with the master equation had been solved recently by Janvresse, by Maslen, and by Carlen, Carvalho and Loss. My argument provides some new estimates about the entropy variant of this famous spectral gap problem.
The fact that my entropy production estimates need the distribution function to be very smooth may look disappointing. Indeed, for the Boltzmann equation with, say, hard spheres and spatial homogeneity, the solution is not smoother than the initial datum. However, together with Clément Mouhot (at the time my PhD student), we proved that the result of fast convergence to equilibrium also applies to rough initial data. We did this by showing that the solution can be decomposed as the sum of a vanishingly small part and a part which is very smooth, uniformly in time. This is related to a lot of nice harmonic analysis about the structure of the Boltzmann collision operator; for more details consult
C. Mouhot and C. Villani: Regularity theory for the spatially homogeneous Boltzmann equation with cut-off. Arch. Rational Mech. Anal. 173, 2 (2004), 169–212. PDF File
Space-inhomogeneous convergence to equilibrium
Around 2000, all explicit results of convergence to equilibrium were dealing with the spatially homogeneous theory. Together with Desvillettes, we set up a program to systematically study the convergence to equilibrium in a spatially inhomogeneous context. “Instantaneous” entropy production estimates are definitely not enough; one should try to estimate at the same time how the distribution function becomes close to the associated local Maxwellian, and how fast this local Maxwellian approaches the global Maxwellian which maximizes the entropy. Information-theoretic estimates and fluid mechanics are closely tied together in this problem, and this in my opinion is one of its most appealing features.
A first step in the program was the study of the linear kinetic Fokker-Planck equation. Although much simpler, this model shares some properties with the Boltzmann equation. We studied the convergence to equilibrium by using only constructive methods based on entropy production, using as little as possible the linear character of the equation. We got a hold on this problem by estimating at the same time the first time-derivative of the entropy functional, and the second time-derivative of the local entropy functional (relative entropy with respect to the local Maxwellian), and then applying two logarithmic Sobolev inequalities (one for the position, one for the velocity) together with a horrible interpolation inequality. In the end we reduced the proof of convergence to equilibrium to the study of a system of two coupled differential inequalities, one of first order and the other of second order, which we estimated in a very pedestrian way (since no maximum principle or Gronwall lemma was available). The result was the following paper:
L. Desvillettes and C. Villani: 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. PDF File
Its main result is a “nonlinear” proof that solutions of the linear kinetic Fokker-Planck equation converge towards equilibrium faster than any inverse power of time. Of course, you would expect that a linear strategy would yield better results, and I advertised for this problem. Surprisingly enough, it seemed that only nonconstructive proofs of convergence had been given for this model. It is only in 2002 that more precise estimates were obtained by Frédéric Hérau and Francis Nier. This was a beautiful contribution, but definitely nontrivial: Functional calculus, spectral estimates for non-sectorial linear operators…
At the same time, Desvillettes and I continued our exploration of convergence to equilibrium for the really tough guy, namely the Boltzmann equation. Soon we realized that the same method which we had used for the linear kinetic Fokker-Planck equation could be applied to the Boltzmann equation if we assumed a lot of regularity from the solutions. However, even under this regularity assumption, the problem was still much more difficult than we expected, and it took us several years to get everything under control. One of the most serious problems came from the fact that the Boltzmann equation has three conservation laws instead of just one. Also the geometric picture was much more complicated and we had to introduce several inequalities of Poincaré-type to control the fluctuations in the position variable. Thus, instead of a system of two differential inequalities combined with three functional inequalities, we ended up with a system of eight differential inequalities combined with eight functional inequalities…. Putting everything together, we finally obtained the result of explicit convergence to equilibrium, again faster than any inverse power of time. Much more explanations can be found in our paper:
L. Desvillettes and C. Villani: On the trend to global equilibrium for spatially inhomogeneous systems: the Boltzmann equation. Invent. Math. 159, 2 (2005), 245–316. PDF File
The regularity assumptions are very stringent, but 1. they are satisfied at least in the close-to-equilibrium regime, as showed by Guo in an important series of works; 1.they can be relaxed by using the tricky regularity theory of the Boltzmann equation. Anyway this is not the point of our study: our goal was to convert regularity bounds into convergence rates. At the time of writing, combined with Guo’s estimates, these results provided the first explicit results of convergence, even in the close-to-equilibrium regime! Since then, a more direct treatment of this perturbative situation was developed by Guo and Strain; but this does not replace the generality of the non-perturbative result in our paper. As a by-product of our study of convergence to equilibrium for the Boltzmann equation, we felt the need to study a certain non-conventional inequality of Korn-type. It looked just as the plain Korn L2 inequality, but the usual zero-displacement boundary condition was replaced by a weaker tangency boundary condition. For the inequality to hold true, additional geometrical conditions had to be imposed on the domain: in dimension 2 or 3, it should not possess any axis of symmetry. Precisely this inequality was needed in our analysis of convergence to equilibrium, and to be consistent it had to be established in a constructive way. We were able to bound the constant in terms of a certain domain functional introduced by Grad thirty years ago, which we called “Grad’s number”. In fact Grad proved a much stronger inequality than the Korn inequality which we were studying, but that much stronger inequality was false; still Grad should be given enormous credit for his beautiful intuition on this topic. To our surprise, we further found that Grad’s number could in turn be estimated in a very simple way in terms of a Monge-Kantorovich minimization problem. To know more about this you can consult
L. Desvillettes and C. Villani: On a variant of Korn’s inequality arising in statistical mechanics. ESAIM Contrôle Optim. Calc. Var 8 (2002), 603-619. PDF File
(Since then, these estimates have been simplified by Figalli.) This set of results about convergence to equilibrium for the Boltzmann equation is described in my 2004 Harold Grad lecture, as well as my lecture notes on entropy production.
- simplify the proof of our main results;
- find general methods applying to various models sharing similar features; 1. find a unified approach to Fokker-Planck and Boltzmann equations, in particular revisit the Hérau-Nier results. This led me to the beginning of a theory of hypocoercivity (the word itself was suggested by Thierry Gallay), which is inspired by, but distinct from, hypoellipticity. With the help of some simple “algebraic” tools I could derive some results of exponential convergence to equilibrium for rather general operators in an abstract “Hörmander form”, under some commutator assumptions, which generalize, simplify and improve the results by Hérau and Nier. The method lends itself well to an “entropic” version, where spectral estimates are replaced by logarithmic Sobolev inequalities (there are some miracles here with the interplay between diffusion operators and chain-rules). Then I could also prove an abstract and general version of the nonlinear results previously obtained with Desvillettes, and I used this new version to treat degenerate cases (e.g. Boltzmann equation with a spherical domain, or diffusive reflection kernel). All these advances are gathered in the following long memoir:
C. Villani: Hypocoercivity. Mem. Amer. Math. Soc. 202 (2009), no. 950. PDF File
The main results of this work, along with motivations, are also summarized in my Proceedings for the 2006 ICM (Download). Hypocoercivity theory has been developing fast since then, and these methods have found applications in the study of models from micromagnetism, fluid mechanics (stability of Oseen vortices), statistical mechanics (models for propagation of heat), with contributions by Gallay-Gallagher-Nier, Liverani-Olla, and others. Dolbeault, Hairer, Hérau, Mouhot, Neumann and others have also found new results in this direction.
As a by-product of this study, I developed a new method for hypoelliptic regularity, applied mainly to the kinetic Fokker-Planck equation. Besides its elementary nature, this method has the advantage to yield global estimates and to work with measure initial data. Unlike the methods of Hörmander and Kohn, it is based on energy estimates, functional and differential inequalities in the style of Nash. This can also be found in the above-mentioned memoir.
Qualitative properties of kinetic models of granular media
Nowadays, granular materials are fashionable; physicists and mathematicians try to study their properties through a great variety of models. I was interested in the fine study of qualitative properties of some kinetic models for granular media. A brief presentation of the subject can be found in the last chapter of my review about collisional kinetic theory, and in my brief survey on the kinetic theory of granular media.
My first result in this field (in collaboration with José Antonio Carrillo and Robert McCann) was a theorem of exponential convergence to equilibrium for a baby model of granular media undergoing diffusion, which had been studied before by several authors. Another simple kinetic model for granular media consists in a nonlinear drift equation without diffusion, which has been introduced long ago by the physicists McNamara and Young, then studied by Benedetto, Caglioti and Pulvirenti. The equilibrium state can easily be shown to be a Dirac mass in velocity space. This situation is common for models with inelastic collisions, and it is often said that the relevant question is not whether the distribution function will approach the Dirac mass, but how it will do so; in particular, many researchers believe that the distribution tends to resemble a particular self-similar solution, sometimes called Homogeneous Cooling State (HCS). This statement, and even the existence of HCS’s, is subject to debate. In the present situation, it is known that there exists an HCS, and it had been proven by Benedetto, Caglioti and Pulvirenti that the distribution function does approach this HCS faster than it approaches the Dirac mass. This seemed to corroborate the physical intuition… However, Emanuele Caglioti and I proved that the gain obtained when replacing the Dirac mass by the HCS is almost negligible: essentially, it cannot be better than a logarithmic correction! This reflects a property of “slow convergence” to equilibrium for a certain rescaled equation. I used to think that the result was purely mathematical, until my friend Alain Barrat told me that he had observed this slow convergence numerically (without being aware of our results). The mathematical proof uses the Monge-Kantorovich distance to take advantage of the nonlinearity of the drift; it can be found in the following paper: E. Caglioti and C. Villani: Homogeneous cooling states are not always good approximations to granular flows. Arch. Rational Mech. Anal. 163, 4 (2002), 329-343. PDF File
Then I studied another kinetic model together with Irene Gamba and Vladislas Panferov: this is a spatially homogeneous inelastic Boltzmann equation with extra diffusion in velocity space. The model had been previously studied heuristically by some physicists, in particular Matthieu Ernst. By a simple dimensional argument, he suggested that there may exist equilibrium distributions which would be “overpopulated”, in the sense that the distribution tails for large velocities would be much thicker than the distribution tails of a Maxwellian distribution: an equilibrium distribution would look asymptotically like the inverse exponential of a power 3/2 (instead of 2) of the velocity. By applying the well-developed machinery of the spatially homogeneous theory and some new estimates, we were able to prove that the Cauchy problem for this equation admits a unique solution, which becomes instantaneously very regular: not only is it infinitely smooth, but also it has finite moments of all orders, and all these bounds are uniform in time. And we did prove the existence of these overpopulated equilibria, bounded below by an inverse exponential of a 3/2 power as predicted; in fact we proved the stronger result that the solution is overpopulated for all positive times.
I. Gamba, V. Panferov and C. Villani: On the Boltzmann equation for diffusively excited granular media. Comm. Math. Phys. 246, 3 (2004), 503-541. PDF File
To strengthen these results, we are still striving to prove that the equilibria are also controlled from above by a stretched exponential with the same exponent. For this goal we introduced and proved a new maximum principle theorem for the Boltzmann equation. Yet still some work remains to be done until we can reach the desired conclusion. Meanwhile, this new maximum principle allowed us to solve an old irritating problem in the theory of the spatially homogeneous Boltzmann equation, namely the propagation (up to constants) of upper Maxwellian bounds:
I. Gamba, V. Panferov and C. Villani: Upper Maxwellian bounds for the spatially homogeneous Boltzmann equation. Arch. Rational Mech. Anal. 194 (2009), 1, 253–282. PDF File
Collisionless relaxation (Landau damping)
The famous Landau damping predicts the collisionless stability of spatial homogeneity, on relatively short time-scales. This should be compared with the collisionwise stability of Maxwellian distributions, on larger time-scales. The time-scale separation is important both in theory and practice; in classical astrophysics, collisionless phenomena are needed to explain the surprisingly short relaxation time scales of galaxies.
Landau damping was discovered in 1946, and widely studied since then (a review paper from ten 1998 estimated that one third of all research papers in plasma physics directly allude to Landau damping). But theoretical studies only covered the linear level (rigorously) or quasi-linear level (not rigorously). These regimes deal with time scales that are not so large! Already fifty years ago, Backus noticed this problem and cast doubt on the whole linearization procedure; his objection still applies in the quasilinear regime, and had remained unanswered.
In a joint work with Clément Mouhot, we establish nonlinear Landau damping in infinite time, for any interaction no more singular than Coulomb or Newton. The limit case of Coulomb/Newton interaction is handled with specific technical effort. As a corollary, we obtain new stability results for homogeneous equilibria of the Vlasov-Poisson equation, under analytic perturbations.
Some of the new physical insights are:
- Under adequate assumptions, there is indeed convergence in large times for the full nonlinear equation; this was not really expected in the physical literature, since the quasilinear theory of plasmas predicts convergence only after statistical averaging.
- The regularity of the interaction plays a crucial role; in our case the Coulomb/Newton singularity is the limit which we can handle.
- There is a deep link between Landau damping and plasma echo and KAM theory. Roughly speaking, linear Vlasov is a completely integrable system, and its perturbation by the nonlinear interaction leaves the mixing property unaffected (KAM spirit, although there is no small measure set to exclude); and the stability over very long times is made possible by the echo nature of the response.
- Homoclinic/heteroclinic trajectories of the nonlinear Vlasov equation are so many that they fill up a whole neighborhood of a stable homogeneous profile satisfying appropriate conditions. Only infinite dimension can allow such a behavior.
The damping phenomenon is reinterpreted in terms of exchanges of regularity between spatial and kinetic modes, rather than energy; it is driven by the phase mixing mechanism associated with the trajectories of particles. Some of our new tools are the introduction of families of analytic norms (with many indices…), measuring regularity by comparison to solutions of the free transport equation; distinctive functional inequalities; a control of nonlinear echoes; sharp scattering estimates in analytic regularity; and a Newton approximation scheme, whose extremely fast convergence is fully exploited.
The text can be found here:
C. Mouhot and C. Villani. On Landau damping. Acta Mathematica 207, 1 (2011), 29–201. PDF File
…But it is rather substantial! Its writing was actually one of the most intense experiences of my professional life. You may prefer to start with the syntheses texts which I wrote on Landau damping to summarize our work for both physicists and mathematicians.
This work was adapted by Bedrossian and Masmoudi to establish the nonlinear inviscid damping of a perfect fluid, for perturbations of the Couette flow — a problem posed by Orr more than a century ago. While quite similar in spirit to our result of nonlinear Landau damping, the theorem of Bedrossian-Masmoudi is also based on new ideas and crucial technical improvements: paraproducts, refined norms…
A survey article, written for an audience of physicists, summarizes these new developments and puts them into perspective with other results about the large time behavior of classical systems.
Optimal transport, Ricci curvature and functional inequalities with a geometric content
In the mid-nineties, Richard Jordan, David Kinderlehrer and Felix Otto made the beautiful discovery that the heat equation, set in the space of probability measures, is the gradient flow of the negative entropy functional, with respect to a metric structure induced by the quadratic Wasserstein distance. Otto also developed these ideas for some nonlinear diffusion equations of porous medium type, thereby connecting the theory of these equations with the theory of the Monge-Kantorovich minimization problem. A lot of information on this approach can be found in my CIME lecture notes or in my first book on optimal transport.
Around 1998, Otto and I were trying to use the optimal transport formalism to shed a new light on certain classes of functional inequalities of Sobolev type, in particular the logarithmic Sobolev inequalities. Even if the point of view developed by Otto was at the time purely formal, his study of convergence to equilibrium for the porous medium equations had demonstrated that it could help intuition a lot. We found out that the idea of gradient flow with respect to a Monge-Kantorovich structure could lead to a crystal-clear (in our opinion) interpretation of logarithmic Sobolev inequalities; and a simple, optimal-transportation-based new proof of the famous Bakry-Émery theorem.
On the occasion of this study we introduced what we called “HWI inequalities”, which are interpolation-like inequalities involving the entropy or H functional, the quadratic Wasserstein distance W and the Fisher information I. Besides their own interest (they have been applied in spin systems, or hydrodynamic limits of stochastic particle systems), these inequalities contained the Bakry-Émery theorem. In addition we proved that a logarithmic Sobolev inequality always implied the domination of the quadratic Wasserstein distance by the entropy, a functional inequality which we called “Talagrand inequality” because it had been first proven by Talagrand in the particular case of Gaussian reference density. In fact the Talagrand inequality could be interpreted as some kind of dual version of the logarithmic Sobolev inequality. All this can be found in the following paper:
F. Otto and C. Villani: Generalization of an inequality by Talagrand, viewed as a consequence of the logarithmic Sobolev inequality. J. Funct. Anal. 173, 2 (2000), 361-400. PDF File
The methods and insights developed in the paper were probably more important than the results themselves, and this is actually my most quoted paper so far. Since then, many authors have worked to develop this line of research (Bobkov, Cordero-Erausquin, Djellout, Gangbo, Gentil, Gozlan, Guillin, Houdré, Ledoux, Malrieu, Wang, Wu… ) In particular, two alternative proofs of our main results were found out: the first one is due to Bobkov-Gentil-Ledoux, who discovered on this occasion the phenomenon of “hypercontractivity of Hamilton-Jacobi equations”; the second one, amazingly elegant and general, is due to Gozlan and relies on Sanov’s large deviation theorem. Much more information, and applications, are reviewed in Chapters 8 and 9 in my first book on optimal transport, and in Chapter 22 of my second book on optimal transport.
In my previous work with Otto, we had given heuristic arguments in favor of the following conjecture. Take a smooth Riemannian manifold, and for any probability density on it, define the H functional, or negative of the entropy, by the usual Boltzmann-Shannon formula. Then this functional is displacement convex, in the sense that it is convex along geodesics of optimal transport, as soon as the Ricci curvature is nonnegative everywhere. This conjecture was rigorously proven by Cordero-Erausquin, McCann and Schmuckenschläger. It emerged later that not only does this property characterize the nonnegativity of Ricci curvature, but also it can be used to encode Ricci bounds in an efficient, versatile and stable way. So this opened the door to a quite unexpected synthetic treatment of Ricci curvature bounds in terms of optimal transport tools. The field was pioneered by Lott, Sturm, Von Renesse and myself. In two independent and complementary contributions, Lott and I on one hand, Sturm on the other hand, examined the possibility to develop a theory of generalized lower bounds for the Ricci curvature in metric-measure spaces. Our main results are that this new formulation is
- more general than the usual formulation since it allows for nonsmooth geometric spaces;
- stable under measured Gromov-Hausdorff convergence; and
- usable to prove nonsmooth versions of Riemannian theorems (Bishop-Gromov, Poincaré, log Sobolev, Bonnet-Myers, etc.) Our paper with Lott can be found here:
J. Lott and C. Villani. Ricci curvature for metric-measure spaces via optimal transport. Ann. of Math. 169, 3 (2009), 903-991. PDF File
This work and the other one by Sturm were the starting point for a number of developments, which are reviewed in Part III of my book, Optimal transport, old and new.
Optimal transport and isoperimetric/Sobolev inequalities
Around 2002, Manuel Del Pino and Jean Dolbeault made the connection between a class of optimal Gagliardo-Nirenberg interpolation inequalities and the problem of convergence to equilibrium for porous-medium type equations. Being aware with the links of optimal transport to these equations, I soon convinced myself that there should be a direct proof of these Gagliardo-Nirenberg inequalities by optimal transport. Together with Dario Cordero-Erausquin and Bruno Nazaret, we found this connection and obtained the desired new, simple and direct proof; at the same time it provided a new straightforward proof of the optimal Sobolev inequalities. Our argument applied with hardly any modification to arbitrary norms, and was sufficiently robust that we could treat cases of equality as well (with a little help from Almut Burchard). The relation of transport to isoperimetry, and therefore to L1 Sobolev inequalities, had been known for two decades since the work of Gromov; but a direct connection to Lp Sobolev inequalities, for any value of p, was nowhere to be seen before our work. All this can be found in
D. Cordero-Erausquin, B. Nazaret and C. Villani: A new approach to sharp Sobolev and Gagliardo-Nirenberg inequalities. Adv. Math. 182, 2 (2004), 307-332. PDF File
I further pushed the study of isoperimetry and optimal Sobolev inequalities in a collaboration with Francesco Maggi. The method worked beyond our expectations: not only were we able to prove a “very sharp” trace Sobolev inequality conjectured by Brézis and Lieb twenty years ago, but in addition we could extend the inequalities to all values of p (not just p=2) and all norms, Euclidean or not. This was included in a “fully nonparametric inequality” which also contained at the same time the usual isoperimetric inequality and the optimal Sobolev inequality.
F. Maggi and C. Villani. Balls have the worst Sobolev inequalities. J. Geom. Anal. 15, 1 (2005), 83-121. PDF File
The method was even more developed by later works in which I did not participate; I will quote in particular an amazingly sharp quantitative isoperimetric inequality for general norms (with an explicit sharp remainder term) by Figalli, Maggi and Pratelli.
Regularity of optimal transport in curved geometry
The regularity of the Monge–Kantorovich optimal transport map has become a fashionable and fascinating topic in fully nonlinear partial differential equations. In the nineties, the main results in the case of a quadratic cost function in Euclidean space were obtained by Caffarelli and Urbas. For ten years or so, other geometries remained untouched.
Then Ma-Trudinger-Wang and Loeper revolutionized the field. The first set of authors introduced a certain differential condition, of order 4 in the cost function, which enables the development of certain regularity estimates. Conversely, Loeper showed that at least a weakened version of this condition is necessary to have any decent regularity theory. All this is reviewed in Chapter 12 of my book Optimal Transport, old and new.
These works were mainly concerned with cost functions defined on product domains, or with the squared geodesic distance on the sphere, for which the cut loci are extremely simple. For more general Riemannian manifolds, the cut locus entails important additional difficulties, and a rich interplay between smooth and nonsmooth Riemannian geometry, which I started to explore together with Alessio Figalli, Grégoire Loeper and Ludovic Rifford. As a first step, Loeper and I considered the simple case when the cut locus is nonfocal. Under these assumptions, and a strong form of the Ma-Trudinger-Wang condition, we adapted the whole of the regularity theory of Loeper and Ma-Trudinger-Wang. The most important intermediate result consisted in a striking geometric property of the cut locus: under these assumptions, the manifold appears convex in each exponential chart.
G. Loeper and C. Villani: Regularity of optimal transport in curved geometry: the nonfocal case. Duke Math. J. 151 (2010), 431–485. PDF File
The stability of the Ma-Trudinger-Wang condition (either under limit or under perturbation) is a very tricky issue, once again because of the focalization phenomenon. It is not even a priori clear that the condition is stable under C4 convergence. However, using synthetic reformulations of the condition, I could prove its stability under the much weaker notion of Gromov-Hausdorff condition. This result may be even more striking than the stability result for Ricci curvature, because it implicitly involves derivatives of the metric up to the fourth order! Using related techniques, I could also obtain the stability of “mesoscopic Hölder regularity” in the C2 topology. Details, and precise statements, can be found in the following text:
C. Villani: Stability of a fourth-order curvature condition arising in optimal transport theory. J. Funct. Anal. 255, 9 (2008), 2683-2708. PDF File
In a new set of results in collaboration with Alessio Figalli and Ludovic Rifford, we prove that for surfaces, the strict version of the Ma-Trudinger-Wang condition is stable under perturbation in the C4 topology, if the nonfocal domain at any point (that is, the region enclosed by the curve of first focal time of the exponential map in the tangent space) is uniformly convex. We also show that a “flat enough” axisymmetric ellipsoid does not satisfy the Ma-Trudinger-Wang condition; so optimal transport is not necessarily smooth on an ellipsoid, while it is smooth on the sphere! This is found in the following paper:
A. Figalli, L. Rifford and C. Villani. On the Ma-Trudinger-Wang tensor on surfaces. Calc. Var. Partial Differential Equations 39, 3-4 (2010), 307–332. PDF File
A notable outcome of this program was some new understanding of the global geometry of the cut locus, a subject which had made not so much progress since Klingenberg’s fifty years old results. For instance, we prove the following “convex Earth” theorem: take a smooth perturbation of the round metric on the n-dimensional sphere; then all injectivity domains of the exponential maps are uniformly convex. Stated otherwise: take an almost round sphere, choose an origin and draw a chart by means of exponential coordinates; then the domain of the chart is uniformly convex.
A. Figalli, L. Rifford and C. Villani. Nearly round spheres look convex. Amer. J. Math. 134, 1 (2012), 109-139. PDF File
Combining these contributions, we show that on surfaces, the convexity of the nonfocal domains implies the semiconvexity of injectivity domains; whether that semiconvexity property is true in general has been an open problem for some time already.
A. Figalli, L. Rifford and C. Villani. Tangent cut loci on surfaces. Differential Geom. Appl. 29, 2 (2011), 154-159. PDF File
In a further paper we clarify the relation between the regularity of optimal transport and the geometry, showing that the MTW condition and the convexity of injectivity domains are essentially necessary and sufficient to ensure the continuity of optimal transport between arbitrary smooth positive probability densities; in dimension 2 one can formulate this as a necessary and sufficient condition.
A. Figalli, L. Rifford and C. Villani. Necessary and sufficient conditions for continuity of optimal transport maps on Riemannian manifolds. PDF File