Solutions globales pour des \'equations de Schr\"odinger sur-critiques en toutes dimensions

Abstract

In \cite{poiret}, we explain how we can construct global solutions for the cubic Schr\"odinger equation in three dimensional with initial data in $ L^2(\mathds{R}^3) $. The main ingredient of this proof is the existence of the bilinear estimate for the harmonic oscillator. This estimate is true in all dimensions $ d \geq 2 $ and we can adapt the proof in these cases. We explain in this paper how we can use the smoothing effect to obtain an analogous theorem in all dimensions, in particular in dimension 1. The gain of regularity is lower but we can choose any basis of eigenfunctions and random variables more general than Gaussian.