Improved global well-posedness for mass-critical nonlinear Schr\"odinger equations on tori

TL;DR

This paper establishes new global well-posedness results for mass-critical nonlinear Schrödinger equations on one- and two-dimensional tori, extending the range of initial data regularity for which solutions exist globally.

Contribution

It improves the known regularity thresholds for global well-posedness of mass-critical NLS on tori in low dimensions, including irrational tori, with distinctions between focusing and defocusing cases.

Findings

Global well-posedness for s > 1/3 on the circle with small L^2 norm.

Global well-posedness for s > 3/5 on two-dimensional tori.

Different conditions for focusing and defocusing nonlinearities.

Abstract

We show new global well-posedness results for mass-critical nonlinear Schr\"odinger equations on tori in one and two dimensions. For the quintic nonlinear Schr\"odinger equation on the circle we show global well-posedness for initial data in $H^s(\mathbb{T})$ for $s>\frac{1}{3}$ and $\| u_0 \|_{L^2(\mathbb{T})} \ll 1$. In two dimensions we show global well-posedness on possibly irrational tori for $s>\frac{3}{5}$. In the focusing case we need a smallness condition for the mass, whereas in the defocusing case large mass is covered.