Probabilistic global well-posedness for the supercritical nonlinear harmonic oscillator

Aurélien Poiret (1) , Didier Robert (2) , Laurent Thomann (2)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 LMJL - Laboratoire de Mathématiques Jean Leray

Thanks to an approach inspired from Burq-Lebeau, we prove stochastic versions of Strichartz estimates for Schrödinger with harmonic potential. As a consequence, we show that the nonlinear Schrödinger equation with quadratic potential and any polynomial nonlinearity is almost surely locally well-posed in L^2(R^d) for any d≥2. Then, we show that we can combine this result with the high-low frequency decomposition method of Bourgain to prove a.s. global well-posedness results for the cubic equation: when d=2, we prove global well-posedness in H^s(R^{2}) (the Sobolev space based on the harmonic oscillator) for any s>0, and when d=3 we prove global well-posedness in H^s(R^3) for any s>1/6, which is a supercritical regime.

Mots clés: Harmonic oscillator; supercritical non-linear Schrödinger equation; random initial conditions; scattering; global solutions

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]; Analyse classique [math.CA]; Théorie spectrale [math.SP]

Aurélien Poiret, Didier Robert, Laurent Thomann. Probabilistic global well-posedness for the supercritical nonlinear harmonic oscillator. 2013. ⟨hal-00857679v1⟩