Probabilistic global well-posedness for the supercritical nonlinear harmonic oscillator
type: Distribution
access URL: https://hal.science/hal-00857679
format: HTML
issued: 2026-05-05T11:48:33.844352
modified: 2026-05-05T11:48:33.829432

test_moissonnage_selune
type: Agent

Probabilistic global well-posedness for the supercritical nonlinear harmonic oscillator
type: Dataset
contact point: Organization
  fn: CCSD
keyword: 35q55--35r60--35p05
keyword: harmonic-oscillator
keyword: mathmath-mpmathematics-mathmathematical-physics-math-ph
keyword: mathmath-spmathematics-mathspectral-theory-mathsp
keyword: supercritical-non-linear-schrodinger-equation
keyword: global-solutions
keyword: scattering
keyword: journal-articles
keyword: mathmath-apmathematics-mathanalysis-of-pdes-mathap
keyword: random-initial-conditions
keyword: mathmath-camathematics-mathclassical-analysis-and-odes-mathca
keyword: infoeu-reposemanticsarticle
landing page: 2157-5045
description: 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 $L^{2}(\R^{d})$ 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.
issued: 2026-05-05T11:48:33.841709
language: en
modified: 2026-05-05T11:48:33.841714