As I promised (but with a lot of delay!), I develop a little bit on the program of the lectures. Even though I did not find time to write lecture notes, I will be glad to give details to anybody interested. Some of the topics below are developed in my research page, together with preprints. Note however that in lectures 1, 2 and 4 some ``very new'' results were presented, which are not yet in preprint form. Because of the important amount of material concerning regularity estimates in the non-cut-off case, I may in fact decide to publish some of it directly in book form.
I started this lecture with some motivation for the study of qualitative properties of the Boltzmann equation. From the theoretical point of view, a very strong motivation is the desire to put the maximum entropy principle on sound mathematical basis, starting from a many-particle system.
This first lecture was mainly devoted to properties of the Boltzmann collision kernel, and the spatially homogeneous Boltzmann equation. I reviewed the existing theory, insisting on three types of estimates: Sobolev regularity, moment bounds and quantitative lower bounds for the solution. I laid particular emphasis on two phenomena: propagation of regularity and exponential damping of singularities in the cut-off case (as quantified in my recent work with Clément Mouhot), and appearance of regularity in the non-cut-off case.
Three new results, not yet published, were explained. The first is the propagation of upper Maxwellian bound, derived jointly with Vladislas Panferov for the spatially homogeneous Boltzmann equation with cut-off. The second one is a new, quantitative lower bound estimate for the spatially homogeneous Boltzmann equation without cut-off, in the form of a stretched Maxwellian. On that last topic, Clément Mouhot has recently obtained independent results. Finally, I sketched the proof of a monsterly technical estimate on the Boltzmann collision operator without cut-off, showing that it is not more singular than a certain optimal fractional power of the Laplace operator.
On the whole, this first lecture was certainly too ambitious and Pierre-Louis Lions did not hesitate to call me ``merciless''. I tried to make the contents of the next lectures lighter.
This lecture was devoted to known results of regularity in the context of the spatially inhomogeneous Boltzmann equation. I explained a recent conditional regularity result for the Boltzmann equation without cut-off: whenever a solution satisfies the following three a priori estimates: (i) density of particles (in space) bounded from below (in particular no vacuum), (ii) density of kinetic energy bounded from above, and (iii) pressure matrix tensor field uniformly positive definite, then this solution is infinitely smooth for positive times.
I explained how such results should be considered in the general picture of hypoelliptic estimates, and gave a brief history of the study of such regularization effects: in particular, Hörmander's commutator method, and the velocity-averaging method. I discussed Bouchut's recent regularity results for the kinetic Fokker-Planck equation. Finally, I explained a new proof of Hörmander's regularity theorem in a kinetic context, by a semigroup argument which yields almost optimal exponents, and in some cases uniform in time estimates.
I started this lecture by announcing the birth of my daughter Aëlle, that very morning. I praised the TGV for giving me the opportunity to attend the birth in the morning and lecturing in the afternoon.
Then I started the study of convergence to equilibrium for the Boltzmann equation. I reviewed known results in the spatially homogeneous setting, and discussed in detail my recent results about entropy production and Cercignani's conjecture. After giving a complete proof of this conjecture in the case of (nonrealistic) overquadratic collision kernel, I explained how essentially sharp entropy-entropy production inequalities could be derived for realistic collision kernels via ad hoc non-concentration inequalities. This discussion also encompassed quantitative estimates of the spectral gap for Boltzmann's linearized equation, as obtained recently by Céline Baranger and Clément Mouhot.
The last lecture was entirely devoted to the convergence to equilibrium for spatially inhomogeneous kinetic-like equations. I started with a basic example, the kinetic Fokker-Planck equation: how to obtain exponential convergence rates? I emphasized the analogy with the hypoelliptic problematic, and reviewed existing probabilistic methods based on Markov chains theory (works of Stuart, Mattingly, Talay, Rey-Bellet, etc.) and analytical methods based on pseudo-differential operators in the style of Kohn's treatment of the hypoellipticity (Hérau and Nier, Eckmann and Hairer). After that I presented a new result of exponential convergence to equilibrium, which also applies to the kinetic Fokker-Planck equation, but which is expressed in a rather abstract operator-theoretical language, and whose proof is very elementary. [Since then I have developed that method in such a way as to cover many interesting examples, with rather natural assumptions.]
After that I discussed my recent conditional result of convergence to equilibrium for the Boltzmann equation, proven with Laurent Desvillettes. We show that the convergence is faster than any inverse power of time, if there are uniform bounds below on the density, and uniform bounds on the regularity of the solution, at arbitrarily high order. After explaining the main ingredients of the proof, I showed some numerical simulations prepared by Francis Filbet, which seem to strongly confort my hypothesis that the kinetic (as opposed to hydrodynamic) part of the entropy may undergo sharply marked oscillations as time goes on. This phenomenon is clearly a consequence of the spatial inhomogeneity.