Probabilistic global well-posedness for the supercritical nonlinear harmonic oscillator

TL;DR

This paper establishes probabilistic local and global well-posedness for nonlinear Schrödinger equations with harmonic potential, including supercritical regimes, using stochastic Strichartz estimates and frequency decomposition methods.

Contribution

It introduces stochastic Strichartz estimates for Schrödinger with harmonic potential and proves almost sure global well-posedness in supercritical regimes, extending previous deterministic results.

Findings

Almost sure local well-posedness in $L^2( ^d)$ for quadratic potential and polynomial nonlinearities.

Global well-posedness in $H^s$ for cubic NLS in 2D and 3D with supercritical regularity.

Scattering results for $L^2$-supercritical and subcritical equations without potential.

Abstract

Thanks to an approach inspired from Burq-Lebeau \cite{bule}, we prove stochastic versions of Strichartz estimates for Schr\"odinger with harmonic potential. As a consequence, we show that the nonlinear Schr\"odinger equation with quadratic potential and any polynomial non-linearity 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 $s>1/6$, 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.