We show existence of global solutions for the gravity water waves equation in dimension 3, in the case of small data. The proof combines energy estimates, which yield control of L 2 related norms, with dispersive estimates, which give decay in L ∞ . To obtain these dispersive estimates, we use an analysis in Fourier space; the study of space and time resonances is then the crucial point.

We study Sobolev regularity disturbances to the periodic, plane Couette flow in the 3D incompressible Navier-Stokes equations at high Reynolds number Re. Our goal is to estimate how the stability threshold scales in Re: the largest the initial perturbation can be while still resulting in a solution that does not transition away from Couette flow. In this work we prove that initial data which satisfies u in H σ ≤ δRe −3/2 for any σ > 9/2 and some δ = δ(σ) > 0 depending only on σ, is global in time, remains within O(Re −1/2 ) of the Couette flow in L 2 for all time, and converges to the class of "2.5 dimensional" streamwise-independent solutions referred to as streaks for times t Re 1/3 . Numerical experiments performed by Reddy et. al. with "rough" initial data estimated a threshold of ∼ Re −31/20 , which shows very close agreement with our estimate.

Abstract. We show existence of global solutions for the gravity water waves equation in dimension 3, in the case of small data. The proof combines energy estimates, which yield control of L 2 related norms, with dispersive estimates, which give decay in L ∞ . To obtain these dispersive estimates, we use an analysis in Fourier space; the study of space and time resonances is then the crucial point.

We consider the cubic nonlinear Schrödinger (NLS) equation set on a two dimensional box of size L with periodic boundary conditions. By taking the large box limit L → ∞ in the weakly nonlinear regime (characterized by smallness in the critical space), we derive a new equation set on R 2 that approximates the dynamics of the frequency modes. This nonlinear equation turns out to be Hamiltonian and enjoys interesting symmetries, such as its invariance under the Fourier transform, as well as several families of explicit solutions. A large part of this work is devoted to a rigorous approximation result that allows to project the long-time dynamics of the limit equation into that of the cubic NLS equation on a box of finite size.

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Abstract. We study the continuous resonant (CR) equation which was derived in [7] as the large-box limit of the cubic nonlinear Schrödinger equation in the small nonlinearity (or small data) regime. We first show that the system arises in another natural way, as it also corresponds to the resonant cubic Hermite-Schrödinger equation (NLS with harmonic trapping). We then establish that the basis of special Hermite functions is well suited to its analysis, and uncover more of the striking structure of the equation. We study in particular the dynamics on a few invariant subspaces: eigenspaces of the harmonic oscillator, of the rotation operator, and the Bargmann-Fock space. We focus on stationary waves and their stability.

We prove weak-strong uniqueness results for the isentropic compressible Navier-Stokes system on the torus. In other words, we give conditions on a weak solution, such as the ones built up by Lions (Compressible Models, Oxford Science, Oxford, 1998), so that it is unique. It is of fundamental importance since uniqueness of these solutions is not known in general. We present two different methods, one using relative entropy, the other one using an improved Gronwall inequality due to the author; these two approaches yield complementary results. Known weak-strong uniqueness results are improved and classical uniqueness results for this equation follow naturally.

We study dynamical properties of the cubic lowest Landau level equation, which is used in the modeling of fast rotating Bose-Einstein condensates. We obtain bounds on the decay of general stationary solutions. We then provide a classification of stationary waves with a finite number of zeros. Finally, we are able to establish which of these stationary waves are stable, through a variational analysis.2000 Mathematics Subject Classification. 35Q55 ; 37K05 ; 35C07 ; 35B08. 1 2 PATRICK GÉRARD, PIERRE GERMAIN, AND LAURENT THOMANN 7.2. Minimizers of G µ = 8πH + µP for M fixed 30 7.3. Minimizers of P for H and M fixed 33 7.4. Stability of stationary waves with finite mass and a finite number of zeros 37 Appendix A. Some explicit M -stationary waves 38 Appendix B. The dictionary 40 Appendix C. Sobolev spaces 41 References 42