Vol. 7, No. 4, 2014

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Probabilistic global well-posedness for the supercritical nonlinear harmonic oscillator

Aurélien Poiret, Didier Robert and Laurent Thomann

Vol. 7 (2014), No. 4, 997–1026
Abstract

Thanks to an approach inspired by Burq and Lebeau [Ann. Sci. Éc. Norm. Supér. (4) 6:6 (2013)], 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 L2(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 s(2) for any s > 0, and when d = 3 we prove global well-posedness in s(3) for any s > 1 6, which is a supercritical regime.

Furthermore, we also obtain almost sure global well-posedness results with scattering for NLS on d without potential. We prove scattering results for L2-supercritical equations and L2-subcritical equations with initial conditions in L2 without additional decay or regularity assumption.

Keywords
harmonic oscillator, supercritical nonlinear Schrödinger equation, random initial conditions, scattering, global solutions
Mathematical Subject Classification 2010
Primary: 35P05, 35Q55, 35R60
Milestones
Received: 20 September 2013
Revised: 9 April 2014
Accepted: 8 May 2014
Published: 27 August 2014
Authors
Aurélien Poiret
Laboratoire de Mathématiques
UMR 8628 du CNRS
Université Paris Sud
91405 Orsay Cedex
France
Didier Robert
Laboratoire de Mathématiques J. Leray
UMR 6629 du CNRS
Université de Nantes
2, rue de la Houssinière
44322 Nantes Cedex 03
France
Laurent Thomann
Laboratoire de Mathématiques J. Leray
UMR 6629 du CNRS
Université de Nantes
2, rue de la Houssinière
44322 Nantes Cedex 03
France