Probabilistic global well-posedness for the supercritical nonlinear harmonic oscillator

Poiret, Aurélien; Robert, Didier; Thomann, Laurent

doi:10.2140/apde.2014.7.997

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Probabilistic global well-posedness for the supercritical nonlinear harmonic oscillator

Aurélien Poiret, Didier Robert and Laurent Thomann

Vol. 7 (2014), No. 4, 997–1026

DOI: 10.2140/apde.2014.7.997

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 $L^{2} (ℝ^{d})$ for any $d \geq 2$ . 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 $ℋ^{s} (ℝ^{2})$ for any $s > 0$ , and when $d = 3$ we prove global well-posedness in $ℋ^{s} (ℝ^{3})$ for any $s > \frac{1}{6}$ , which is a supercritical regime.

Furthermore, we also obtain almost sure global well-posedness results with scattering for NLS on $ℝ^{d}$ without potential. We prove scattering results for $L^{2}$ -supercritical equations and $L^{2}$ -subcritical equations with initial conditions in $L^{2}$ 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

Vol. 7, No. 4, 2014