Probabilistic global well-posedness for the supercritical nonlinear harmonic oscillator

Abstract

Thanks to an approach inspired from Burq-Lebeau \cite{bule}, 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 non-linearity is almost surely locally well-posed in $L^2({\mathbb{R}}^d)$ 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 $\H^{s}(\R^{2})$ 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 ${\mathbb{R}}^d$ 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.

32 pages. The limit case has been treated in the scattering result in Theorem 1.4. To appear in Analysis \& PDE

References in corpus (4)

• Random data Cauchy theory for supercritical wave equations I: Local theory
• Almost sure well-posedness of the cubic nonlinear Schrödinger equation below L^2(T)
• Random data Cauchy theory for supercritical wave equations II : A global existence result
• Continuous dependence for NLS in fractional order spaces

Cited by in corpus (2)

• On the continuous resonant equation for NLS: II. Statistical study
• Refined mass-critical Strichartz estimates for Schrödinger operators