Bilinear Strichartz estimates and almost sure global solutions for the nonlinear Schrödinger equation

2023/04/21 by Nicolas Burq, Burq, Nicolas, Aurélien Poiret +3
Mathematics · #Advanced Mathematical Physics Problems

paper · pdf · doi:10.48550/arxiv.2304.10979

Abstract

The purpose of this article is to construct global solutions, in a probabilistic sense, for the nonlinear Schrödinger equation posed on 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 ℝ³. Then, thanks to the lens transform, we are able to obtain global in time solutions for the nonlinear Schrödinger equation without harmonic potential. Secondly, we prove a Kato type smoothing estimate for the linear Schrödinger equation with harmonic potential. This allows us to consider the Schrödinger equation with a nonlinearity of odd degree in a supercritical regime, in any dimension d≥ 2.

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