The Euler--Poisson system in 2D: global stability of the constant equilibrium solution

We consider the (repulsive) Euler-Poisson system for the electrons in two dimensions and prove that small smooth perturbations of a constant background exist for all time and remain smooth (never develop shocks). This extends to 2D the work of Guo.Comment: 39 page

Global regularity for the energy-critical NLS on S3\mathbb{S}^3

We establish global existence for the energy-critical nonlinear Schr\"odinger equation on $\mathbb{S}^3$. This follows similar lines to the work on $\mathbb{T}^3$ but requires new extinction results for linear solutions and bounds on the first nonlinear iterate at a Euclidean profile that are adapted to the new geometry.Comment: to appear in the Annales IHP, Analyse non lineaire. arXiv admin note: text overlap with arXiv:1102.5771, arXiv:1101.452

Topography influence on the Lake equations in bounded domains

We investigate the influence of the topography on the lake equations which describe the two-dimensional horizontal velocity of a three-dimensional incompressible flow. We show that the lake equations are structurally stable under Hausdorff approximations of the fluid domain and $L^p$ perturbations of the depth. As a byproduct, we obtain the existence of a weak solution to the lake equations in the case of singular domains and rough bottoms. Our result thus extends earlier works by Bresch and M\'etivier treating the lake equations with a fixed topography and by G\'erard-Varet and Lacave treating the Euler equations in singular domains

The Euler--Maxwell system for electrons: global solutions in 2D2D

A basic model for describing plasma dynamics is given by the Euler-Maxwell system, in which compressible ion and electron fluids interact with their own self-consistent electromagnetic field. In this paper we consider the "one-fluid" Euler--Maxwell model for electrons, in 2 spatial dimensions, and prove global stability of a constant neutral background.Comment: Revised versio