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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 ∞ estimates. The L 2 bounds are proved in the companion paper [5] 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 [5], allows us to deduce our main global existence result.

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Soit v une solution de l'équation de Klein-Gordon quasi linéaire en dimension 1 d'espace v + v = F (v, ∂tv, ∂xv, ∂t∂xv, ∂ 2 x v) à données de Cauchy régulières à support compact, de taille ε → 0. Supposons que F s'annule au moins à l'ordre 2 en 0. On sait alors que la solution v existe sur un intervalle de temps de longueur supérieure ou égale à e c/ε 2 pour une constante positive c, et que pour une nonlinéarité générale F elle explose en temps fini de l'ordre de e c /ε 2 (c > 0). Nous avons conjecturé dans [7] une condition nécessaire et suffisante sur F sous laquelle la solution devrait exister globalement en temps, pour ε assez petit. Nous prouvons dans cet article la suffisance de cette condition. De plus, nous obtenons le premier terme d'un développement asymptotique de v lorsque t → +∞.  2001 Éditions scientifiques et médicales Elsevier SAS ABSTRACT.-Let v be a solution to a quasilinear Klein-Gordon equation in one space dimension v + v = F (v, ∂tv, ∂xv, ∂t∂xv, ∂ 2 x v) with smooth compactly supported Cauchy data of size ε → 0. Assume that F vanishes at least at order 2 at 0. It is known that the solution v exists over an interval of time of length larger than e c/ε 2 for a positive c, and that for a general F it blows up in finite time e c /ε 2 (c > 0). We conjectured in [7] a necessary and sufficient condition on F under which the solution should exist globally in time for small enough ε. We prove in this paper the sufficiency of that condition. Moreover, we get a one term asymptotic expansion for v when t → +∞.  2001 Éditions scientifiques et médicales Elsevier SAS

This paper is devoted to the proof of almost global existence results for Klein-Gordon equations on Zoll manifolds (e.g. spheres of arbitrary dimension) with Hamiltonian nonlinearities, when the Cauchy data are smooth and small. The proof relies on Birkhoff normal form methods and on the specific distribution of eigenvalues of the laplacian perturbed by a potential on Zoll manifolds.

Abstract. The three bilinearities uv, uv, uv for functions u, v : R 2 ×[0, T ] −→ C are sharply estimated in function spaces X s,b associated to the Schrödinger operator i∂t +∆. These bilinear estimates imply local wellposedness results for Schrödinger equations with quadratic nonlinearity. Improved bounds on the growth of spatial Sobolev norms of finite energy global-in-time and blow-up solutions of the cubic nonlinear Schrödinger equation (and certain generalizations) are also obtained.

Consider a quasi-linear system of two Klein-Gordon equations with masses m 1 ; m 2 : We prove that when m 1 a2m 2 and m 2 a2m 1 ; such a system has global solutions for small, smooth, compactly supported Cauchy data. This extends a result proved by Sunagawa (J. Differential Equations 192 (2) (2003) 308) in the semi-linear case. Moreover, we show that global existence holds true also when m 1 ¼ 2m 2 and a convenient null condition is satisfied by the nonlinearities. r 2004 Elsevier Inc. All rights reserved.