Bootstrapped Morawetz estimates and resonant decomposition for low regularity global solutions of cubic NLS on $R^2$

Abstract

We prove global well-posedness for the $L^2$-critical cubic defocusing nonlinear Schrödinger equation on $R^2$ with data $u_0 \in H^s(R^2)$ for $ s > \frac{1}{3}$. The proof combines <em>a priori</em> Morawetz estimates obtained in [4] and the improved almost conservation law obtained in [6]. There are two technical difficulties. The first one is to estimate the variation of the improved almost conservation law on intervals given in terms of Strichartz spaces rather than in terms of $X^{s,b}$ spaces. The second one is to control the error of the <em>a priori</em> Morawetz estimates on an arbitrary large time interval, which is performed by a bootstrap via a double layer in time decomposition.

Locations

• Communications on Pure &amp Applied Analysis - View - PDF
• arXiv (Cornell University) - View - PDF
• TSpace (University of Toronto) - View - PDF