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Today, I give a Graduate Student Seminar lecture whose goal is to introduce to the first and second year graduate students at Penn State a few topics of research in Fluid Dynamics. There are many recent exciting developments in the field, which I only have time to present a few (many students haven’t taken any PDE course!). You may find the slides of my lecture here (up to many details delivered on the board!). You may also enjoy my similar lecture on Kinetic Theory of Gases, also aiming at first and second year students.

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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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I’ve just submitted this paper with Grenier (ENS Lyon) which studies Prandtl’s boundary layer asymptotic expansions for incompressible fluids on the half-space in the inviscid limit. In 1904, Prandtl introduced his well known boundary layers in order to describe the transition from Navier-Stokes to Euler equations in the inviscid limit.

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Emmanuel Grenier and I have just submitted this 84-page! long paper, also posted on arxiv (arXiv:1705.05323). This work is a continuation and completion of the program (initiated in Grenier-Toan1 and Grenier-Toan2) to derive pointwise estimates on the Green function and sharp bounds on the semigroup of linearized Navier-Stokes around a stationary boundary layer profile.

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