Bilinear Strichartz estimates and almost sure global solutions for the nonlinear Schr{ö}dinger equation

Abstract

The purpose of this article is to construct global solutions, in a probabilistic sense, for the nonlinear Schr{ö}dinger equation posed on ${\mathbb{R}}^d$, in a supercritical regime. Firstly, we establish Bourgain type bilinear estimates for the harmonic oscillator which yields a gain of half a derivative in space for the local theory with randomised initial conditions, for the cubic equation in ${\mathbb{R}}^3$. Then, thanks to the lens transform, we are able to obtain global in time solutions for the nonlinear Schr{ö}dinger equation without harmonic potential. Secondly, we prove a Kato type smoothing estimate for the linear Schr{ö}dinger equation with harmonic potential. This allows us to consider the Schr{ö}dinger equation with a nonlinearity of odd degree in a supercritical regime, in any dimension $d\ge 2$.

88 pages