Global solutions and asymptotic behavior for two dimensional gravity water waves

This paper is devoted to the proof of a global existence result for the water waves equation with smooth, small, and decaying at infinity Cauchy data. We obtain moreover an asymptotic description in physical coordinates of the solution, which shows that modified scattering holds. The proof is based on a bootstrap argument involving $L^2$ and $L^\infty$ estimates. The $L^2$ bounds are proved in a companion paper of this article. They rely on a normal forms paradifferential method allowing one to obtain energy estimates on the Eulerian formulation of the water waves equation. We give here the proof of the uniform bounds, interpreting the equation in a semi-classical way, and combining Klainerman vector fields with the description of the solution in terms of semi-classical lagrangian distributions. This, together with the $L^2$ estimates of the companion paper, allows us to deduce our main global existence result.Comment: 100 pages. Our previous preprint arXiv:1305.4090v1 is now splitted into two parts. This is the first one (which has the same title

Periodic solutions of non-linear Schrödinger equations: A para-differential approach

40 pages. This is the version of the paper to appear in Analysis and PDEsThis paper is devoted to the construction of periodic solutions of non-linear Schrödinger equations on the torus, for a large set of frequencies. Usual proofs of such results rely on the use of Nash-Moser methods. Our approach avoids this, exploiting the possibility of reducing, through para-differential conjugation, the equation under study to an equivalent form for which periodic solutions may be constructed by a classical iteration scheme

A quasi-linear Birkhoff normal forms method. Application to the quasi-linear Klein-Gordon equation on S^1

Consider a nonlinear Klein-Gordon equation on the unit circle, with small smooth data . A solution u which, for any interger N, may be extended as a smooth solution on a time-interval of length bounded from below by the size of the data raised to power -N , is called an almost global solution. It is known that when the nonlinearity is a polynomial depending only on u, and vanishing at order at least 2 at the origin, any smooth small Cauchy data generate, as soon as the mass parameter in the equation stays outside a subset of zero measure of R_+*, an almost global solution whose Sobolev norms of higher order stay uniformly bounded. The goal of this paper is to extend this result to general Hamiltonian quasi-linear nonlinearities. These are the only Hamiltonian non linearities that depend not only on u, but also on its space derivative. To prove the main theorem, we develop a Birkhoff normal form method for quasi-linear equations

Relations subordonnantes et coordonnantes pour la désambiguïsation du discours.

Une représentation hiérarchique du discours permet, entre autres, de mettre au jour des contraintes liées à l'accessibilité des référents ainsi qu'au rattachement d'un nouveau constituant. L'objectif de la présente étude est de mettre en lumière un nouvel avantage d'une représentation hiérarchique du discours. Nous démontrons que la distinction relations subordonnantes vs. relations coordonnantes permet de réduire l'ambiguïté de discours avec deux connecteurs