Solutions globales pour des équations de Schrödinger sur-critiques en toutes dimensions

Aurélien Poiret (1)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay

In \cite{poiret}, we explain how we can construct global solutions for the cubic Schrödinger 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.

Mots clés: smoothing effect; global solutions; harmonic oscillator; random data; super critical Schrödinger equations

Domaines: Equations aux dérivées partielles [math.AP]

Aurélien Poiret. Solutions globales pour des équations de Schrödinger sur-critiques en toutes dimensions. 2012. ⟨hal-00717824⟩