Probabilistic global well-posedness for the supercritical nonlinear harmonic oscillator

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 $L^2\left({ℝ}^d\right)$ for any $d\ge 2$. 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 ${\mathcal{\mathscr{H}}}^s\left({ℝ}^2\right)$ for any $s>0$, and when $d=3$ we prove global well-posedness in ${\mathcal{\mathscr{H}}}^s\left({ℝ}^3\right)$ 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.