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.
Kinetic theory: global solution to 3D Vlasov-Poisson
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.
Kinetic Theory, chapter 1: classical kinetic models.
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).
Invalidity of Prandtl’s boundary layers
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.
Green function for linearized Navier-Stokes around a boundary layer profile: near critical layers
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.
Sublayer of Prandtl boundary layers
The aim of this paper (arXiv:1705.04672), with E. Grenier, is to investigate the stability of Prandtl boundary layers in the vanishing viscosity limit: 
Sharp bounds on linear semigroup of Navier Stokes with boundary layer norms
I’ve just uploaded this paper, with E. Grenier, on the arXiv (arXiv:1703.00881), entitled Sharp bounds on linear semigroup of Navier Stokes with boundary layer norms, aiming a better understanding of the classical Prandtl’s boundary layers. Indeed, one of the key difficulties in dealing with boundary layers is the creation of (unbounded) vorticity in the inviscid limit.
Green function for linearized Navier-Stokes around boundary layers: away from critical layers
I’ve just submitted this new paper with E. Grenier (ENS de Lyon) on arxiv (scheduled to announce next Tuesday 1:00GMT), in which we construct the Green function for the classical Orr-Sommerfeld equations and derive sharp semigroup bounds for linearized Navier-Stokes equations around a boundary layer profile. This is part of the long program to understand the stability of classical Prandtl’s layers appearing in the inviscid limit of incompressible Navier-Stokes flows.
On the Zakharov’s weak turbulence theory for capillary waves
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}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cbegin%7Baligned%7D+%5Cpartial_tf+%2B+2+%5Cnu+%7Ck%7C%5E2+f+%5C+%3D+%5C+Q%5Bf%5D+%5Cend%7Baligned%7D+&bg=ffffff&fg=000000&s=0 "\displaystyle \begin{aligned} \partial_tf + 2 \nu |k|^2 f \ = \ Q[f] \end{aligned}")
Prandtl’s layer expansions for steady Navier-Stokes
In 1904, Prandtl conjectured that slightly viscous flows can be decomposed into the inviscid flows away from the boundary and a so-called Prandtl’s layer near the boundary. While various instabilities indicate the failure of the conjecture for unsteady flows (for instance, see Grenier 2000), recently with Y. Guo, we are able to prove that the conjecture holds for certain steady Navier-Stokes flows; see our paper which is to appear on Annals of PDEs.
Inviscid limit for Navier-Stokes in a rough domain
In this paper with Gérard-Varet, Lacave, and Rousset, we prove the inviscid limit of Navier-Stokes flows in domains with a rough or oscillating boundary. Precisely, we study the 2D incompressible Navier-Stokes flows with small viscosity 
Graduate student seminar: Kinetic theory of gases
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).
The Maxwell-Boltzmann approximation for ion kinetic modeling
In this short paper with C. Bardos, F. Golse, and R. Sentis, we are aimed to justify the Maxwell-Boltzmann approximation from kinetic models, widely used in the literature for electrons density distribution; namely, the relation
in which 
Uniform lower bound for solutions to a quantum Boltzmann
With Minh-Binh Tran, we establish uniform lower bounds, by a Maxwellian, for positive radial solutions to a Quantum Boltzmann kinetic model for bosons; see arXiv preprint.
Instabilities in the mean field limit
Together with Daniel Han-Kwan, we recently wrote a paper, entitled “Instabilities in the mean field limit”, to be published on the Journal of Statistical Physics (see here for a preprint). I shall explain our main theorem below.
Math 505, Mathematical Fluid Mechanics: Notes 2
I go on with some basic concepts and classical results in fluid dynamics [numbering is in accordance with the previous notes]. Throughout this section, I consider compressible barotropic ideal fluids with the pressure law 
Math 505, Mathematical Fluid Mechanics: Notes 1
This Spring ’16 semester, I am teaching a graduate Math 505 course, whose goal is to introduce the basic concepts and the fundamental mathematical problems in Fluid Mechanics for students both in math and engineering. The difficulty is to assume no background in both fluids and analysis of PDEs from the students. That’s it!
On the spectral instability of parallel shear flows
This short note is to be published as the proceeding of a Laurent Schwartz PDE seminar talk that I gave last May at IHES, announcing our recent results (on channel flows and boundary layers), which provide a complete mathematical proof of the viscous destabilization phenomenon, pointed out by Heisenberg (1924), C.C. Lin and Tollmien (1940s), among other prominent physicists. Precisely, we construct growing modes of the linearized Navier-Stokes equations about general stationary shear flows in a bounded channel (channel flows) or on a half-space (boundary layers), for sufficiently large Reynolds number $R \to \infty$. Such an instability is linked to the emergence of Tollmien-Schlichting waves in describing the early stage of the transition from laminar to turbulent flows. In fact, the material in this note is only the first half of what I spoke on that day, skipping the steady case!
On wellposedness of Prandtl: a contradictory claim?
Yesterday, Nov 17, Xu and Zhang posted a preprint on the ArXiv, entitled “Well-posedness of the Prandtl equation in Sobolev space without monotonicity” (arXiv:1511.04850), claiming to prove what the title says. This immediately causes some concern or possible contrary to what has been known previously! Here, monotonicity is of the horizontal velocity component in the normal direction to the boundary. It’s well-known that monotonicity implies well-posedness of Prandtl (e.g., Oleinik in the 60s; see this previous post for Prandtl equations). It is then first proved by Gerard-Varet and Dormy that without monotonicity, the Prandtl equation is linearly illposed (and some followed-up works on the nonlinear case that I wrote with Gerard-Varet, and then with Guo). Is there a contradictory to what it’s known and this new preprint of Xu and Zhang? The purpose of this blog post is to clarify this.
Stability of a collisionless plasma
What is a plasma? A plasma is an ionized gas that consists of charged particles: positive ions and negative electrons. To describe the dynamics of a plasma, let 
