The Princeton Companion to Applied Mathematics just appeared:
I have just uploaded on arXiv a joint work with Stéphane Mischler on “Kac’s Program in Kinetic Theory”. In this paper, we answer a set of questions raised by Mark Kac in his seminal paper Foundations of kinetic theory (Proc. Third Berkeley Symp. Math. Stat. & Prob., 1956) about the derivation of Boltzmann equations from many-particle jump processes.
UPDATE: We have just uploaded an announcement note, to appear shortly in the Comptes-rendus de l’Académie des Sciences de Paris.
Soon will happen a Conference in memory of Carlo Cercignani at IHP, Paris 9-11 february which I am co-organizing together with François Bolley, Laurent Desvillettes and Silvia Lorenzani. Moreover several works quoted for the Fields medal 2010 of Cédric Villani are directly related and somehow motivated by the so-called Cercignani’s conjecture in kinetic theory.
As a tribute to Carlo Cercignani, one of founder of the modern mathematical kinetic theory and a great scientist, here is a short presentation about Cercignani’s conjecture. More details (including references and latest results about the conjecture) can be found in this review paper.
We have just preprinted together with Maria Pia Gualdani and Stephane Mischler a new paper entitled “Factorization for non-symmetric operators and exponential H-theorem”:
This is part of a long-term project (started with the paper hal-00076709) on how to obtain relaxation estimates in larger spaces than the usual “symmetrization” space for PDEs arising in statistical physics. A general abstract spectral theory is developped in order to enlarge the functional space of decay estimates for semigroups. Then applications are given for Fokker-Planck and linear(ized) Boltzmann equations (homogeneous and inhomogeneous). The main outcome of the paper is the proof of exponential convergence in the H-theorem for the full Boltzmann equation under a priori smoothness conditions. The detailed abstract can be found below.
Comments are welcome.
I have uploaded a new paper entitled “Hypocoercivity for linear kinetic equations conserving mass”, written in collaboration with Jean Dolbeault from Paris-Dauphine and Christian Schmeiser from Wien University:
This paper deals with hypocoercivity, i.e. obtaining (constructive) rates of exponential decay for linear equations writing as a skew-symmetric part + a partially coercive part. Coercivity is degenerated but can be recovered taking advantage of the commutators with the skew-symmetric. Hence the name “hypocoercivity” by analogy with the hypoelliptic theory of Hormander. This is applied to Fokker-Planck and relaxation models in kinetic theory, and we provide the first theory for these models (without any regularity assumption on the initial data), the first unified theory for diffusive and integral collision models, and the first results for kinetic version of the fast diffusion equation (whose Gibbs state does not separate variables).
Together with my collaborator Stéphane Mischler we have uploaded this morning a new paper on the issue of mean-field limit. It provides a new systematic method for proving in a quantitative manner estimates of chaos propagation which measure fluctuations around the limit deterministic mean-field dynamics, and it is applied to collision jump processes as considered by Kac in the 50ies.
— Part III, michaelmas terms 2010 & 2011 —
Chapter 6 (soon!)
Some elements of history of Kinetic Theory (up to 2001) from the webpage of Dave Levermore.
Sources: Some parts of these lecture notes owe a lot to two sets of hand-written lecture notes on kinetic equations written by Laure Saint-Raymond and Laurent Desvillettes, as well as to the lecture notes available online of G. Allaire and F. Golse on transport and diffusion processes. They also inspire from the classical books of R. Glassey, of C. Cercignani, and of C. Cercignani – R. Illner – M. Pulvirenti, as well as from various recent research papers. Many thanks to Otis Chodosh, who typewrote a first latex version of these notes.