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 denote the electrons density, the elementary charge, the electron temperature, and the electric potential….


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 or incompressible ideal fluids with constant density (and hence, the pressure is an unknown function in the incompressible case). To unify…


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! Anyhow,…


On the spectral instability of parallel shear flows

This short note is 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….


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 of fluid…


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 be the (nonnegative) density distribution of ions and electrons, respectively, at time , position , and particle velocity (or momentum) . The dynamics of a plasma is…


Madelung version of Schrödinger: a link between classical and quantum mechanics

This week I am at the Wolfgang Pauli Institute (WPI) in Vienna for the summer school on “Schrödinger equations”. Several interesting talks on or related to Schrödinger, including those of Y. Brenier on Madelung equations, F. Golse on mean field and classical limits of a N-body quantum system, P. Germain on derivation of the kinetic wave…


Grenier’s nonlinear iterative scheme

In his paper [Grenier, CPAM 2000], Grenier introduced a nonlinear iterative scheme to prove the instability of Euler and Prandtl equations. Recently, the scheme is also proved to be decisive in the study of water waves: [Ming-Rousset-Tzvetkov, SIAM J. Math. Anal., 2015], and plasma physics: [Han-Kwan & Hauray, CMP 2015] or my recent paper with…