On the Hasselmann-Zakharov’s weak turbulence equation 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): describing the dynamics of wave density at wavenumber , . Here, denotes the positive coefficient of fluid viscosity, and is the collision term, describing pure resonant three-wave interactions: with , with…


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…


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 , posed on the following rough domain: whose boundary oscillates at wavelength . Here, . The inviscid limit problem of…


Graduate student seminar: Kinetic theory of gases

Next week, I am giving 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…


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 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….