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 for any . 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 , we prove global well-posedness in for any , and when we prove global well-posedness in for any , which is a supercritical regime.
Furthermore, we also obtain almost sure global well-posedness results with scattering for NLS on without potential. We prove scattering results for -supercritical equations and -subcritical equations with initial conditions in without additional decay or regularity assumption.
Citation
Aurélien Poiret. Didier Robert. Laurent Thomann. "Probabilistic global well-posedness for the supercritical nonlinear harmonic oscillator." Anal. PDE 7 (4) 997 - 1026, 2014. https://doi.org/10.2140/apde.2014.7.997
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