On the global well-posedness for the periodic quintic nonlinear Schr\"odinger equation

TL;DR

This paper proves that solutions to the defocusing quintic nonlinear Schrödinger equation on the two-dimensional torus are globally well-posed in the critical Sobolev space, assuming solutions remain bounded.

Contribution

It establishes global well-posedness for the periodic quintic NLS in the critical Sobolev space under boundedness conditions, extending previous local results.

Findings

Global well-posedness under boundedness assumption

Solutions remain controlled in the critical Sobolev space

Extension of local solutions to global solutions

Abstract

In this paper, we consider the initial value problem for the quintic, defocusing nonlinear Schr\"odinger equation on $\Bbb T^2$ with general data in the critical Sobolev space $H^{\frac{1}{2}} (\Bbb T^2)$. We show that if a solution remains bounded in $H^{\frac{1}{2}} (\Bbb T^2)$ in its maximal interval of existence, then the solution is globally well-posed in $\Bbb T^2$.