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## 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}

## 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 ${\nu}$, posed on the following rough domain:

## 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.

Daniel Han-Kwan and I have just submitted a paper entitled: “Nonlinear instability of Vlasov-Maxwell systems in the classical and quasineutral limits”, which is also available on arxiv: arXiv:1506.08537. In this paper, we study the instability of solutions to the relativistic Vlasov-Maxwell systems in two limiting regimes: the classical limit when the speed of light tends to infinity and the quasineutral limit when the Debye length tends to zero. First, in the classical limit ${\varepsilon \rightarrow 0}$, with ${\varepsilon}$ being the inverse of the speed of light, we construct a family of solutions that converge initially polynomially fast to a homogeneous solution ${\mu}$ of Vlasov-Poisson in arbitrarily high Sobolev norms, but become of order one away from ${\mu}$ in arbitrary negative Sobolev norms within time of order ${|\log \varepsilon|}$. Second, we deduce the invalidity of the quasineutral limit in ${L^2}$ in arbitrarily short time.