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We prove low-regularity global well-posedness for the 1d Zakharov system and 3d Klein-Gordon-Schr\"odinger system, which are systems in two variables $u:\mathbb{R}_x^d\times \mathbb{R}_t \to… (More)

We prove an endpoint multilinear estimate for the $X^{s,b}$ spaces associated to the periodic Airy equation. As a consequence we obtain sharp local well-posedness results for periodic generalized KdV… (More)

We study the long-time behaviour of the focusing cubic NLS on $\R$ in the Sobolev norms $H^s$ for $0 < s < 1$. We obtain polynomial growth-type upper bounds on the $H^s$ norms, and also limit any… (More)

AbstractIn this paper we obtain local in time existence and (suitable) uniqueness
and continuous dependence for the KP-I equation for small data
in the intersection of the energy space and a natural… (More)

We prove new interaction Morawetz type (correlation) estimates in one and two dimensions. In dimension two the estimate corresponds to the nonlinear diagonal analogue of Bourgain's bilinear… (More)

We investigate the initial value problem for a defocusing nonlinear Schr\"odingerequation with exponential nonlinearity. We identify subcritical, critical and supercritical regimes in the energy… (More)

We prove that the one-dimensional Schrodinger equation with derivative in the nonlinear term is globally well-posed in H s for s> 2/3, for small L 2 data. The result follows from an application of… (More)

We consider the Cauchy problem for the one-dimensional periodic cubic non- linear Schrodinger equation (NLS) with initial data below L 2 . In particular, we exhibit nonlinear smoothing when the… (More)

was introduced by Zakharov [7] as a model for Langmuir turbulence in a plasma. The wellposedness theory of the Zakharov system has recently been improved. Local wellposedness below the energy space… (More)