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Archive for the ‘Kinetic theory’ Category

This is part of my lecture notes on Kinetic Theory of Gases, taught at Penn State last semester, Fall 2017. In this part, I’d like to introduce this nice Bardos-Degond 1985’s global solutions to the Vlasov-Poisson system. Of course, the global smooth solutions are already constructed, without any restriction on size of initial data (e.g., Pfaffelmoser, Schaeffer ’91; see also the previous lecture), however they give no information on their asymptotic behavior at large time. Now, for initial data that are sufficiently small near zero, Bardos and Degond were able to construct global smooth solutions that decay in large time. To my knowledge, this was the first result where dispersion is rigorously shown for kinetic equations (they appear to be motivated by similar results for nonlinear wave equations where dispersion was (still is) the key to deduce the global behavior at the large time; e.g., Klainerman, Ponce, Shatah, among others, in the early 80s).

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In this note, I briefly explain my recent joint work with D. Han-Kwan (CNRS, Ecole polytechnique) and F. Rousset (Paris-Sud) on the non-relativistic limit of Vlassov-Maxwell. Precisely, we consider the relativistic Vlasov-Maxwell system, modeling the dynamics of electrons with electron density distribution {f(t,x,v)}, which reads

\displaystyle \partial_t f + \hat v \cdot \nabla_x f + (E + \epsilon \hat v \times B)\cdot \nabla_v f = 0

on {\mathbb{T}^3\times \mathbb{R}^3}, with the relativistic velocity {\hat v = v/\sqrt{1+ \epsilon^2 |v|^2}}.

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As an analogue of the previous post dealing with the classical particles, in this post, I shall formally discuss how similar models for quantum particles arrive. These particles behave like a wave and their dynamics is governed by the Schrödinger equation. We start the chapter with some basic quantum mechanics.

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One of the beautiful results in kinetic theory is to construct the global classical solution to the 3D Vlasov-Poisson system. The result is now classical; see, for instance, chapter 4 of Glassey‘s book. However, I feel the result is a bit non-trivial to convey to students and beginners. Would you agree? Anyway, this post is to try to present this classical result, aiming to be as pedagogical as possible, with the original the good, the bad, and the ugly proof of J. Schaeffer ’91.

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This fall of 2017, I teach a graduate topics course on Kinetic Theory of Gases. The idea is to introduce the foundation of kinetic theory starting from classical mechanics (and also, basic quantum mechanics!), to survey some classical results on both collisional and collisionless kinetic models, and to detail a few selected mathematical topics in the field. The materials are based on several books, papers, and online resources, which I shall mention in the text. Periodically, I shall post my lecture notes for the course here on this blog (email me for a full pdf copy, with figures and precise references).

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In this paper with M.-B. Tran, we construct solutions to the following weak turbulence kinetic equation for capillary waves (cf. Hasselmann ’62, Zakharov ’67):

\displaystyle \begin{aligned} \partial_tf + 2 \nu |k|^2 f \ = \ Q[f] \end{aligned}

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Last week, I gave a graduate student seminar, whose purpose is to introduce to first and second year graduate students (at Penn State) an active and beautiful topics of research, and suggest a few possible ideas for students’ presentation later in the semester. Here are slides of my talk, which focuses on Kinetic Theory of Gases, a topics that I will teach as a graduate topics course, next fall (2017).

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