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## Stability of source defects in oscillatory media

Patterns are ubiquitous in nature, and understanding their formation and their dynamical behavior is always challenging and of great interest. Examples include patterns in fluids (e.g., Rayleigh-Benard convection between two flat plates, Taylor-Couette flow between rotating cylinders, surface waves in hydrothermal fluid flows,…), as well as in nonlinear optics, oscillatory chemical reactions and excitable biological media. Many of them arise from linear instabilities of an homogenous equilibrium, having space, time, or space-time periodic coherent structures such as wave trains (spatially periodic travelling waves). In presence of boundaries or defects, complex patterns form and thus break the symmetry or the periodic structures. Below, I shall briefly discuss some defect structures and my recent work on the subject.

## On the non-relativistic limit of Vlasov-Maxwell

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 ${\hat v = v/\sqrt{1+ \epsilon^2 |v|^2}}$.

## 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 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.

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: