Global well-posedness for the defocusing cubic nonlinear Schr\"odinger equation on $\Bbb T^3$

TL;DR

This paper proves that solutions to the defocusing cubic nonlinear Schrödinger equation on the three-dimensional torus are globally well-posed for initial data in the critical Sobolev space, using profile decomposition and concentration-compactness methods.

Contribution

It establishes the global well-posedness for the equation at the critical regularity on $ ext{T}^3$, advancing understanding of nonlinear Schrödinger equations on compact manifolds.

Findings

Proved global well-posedness for initial data in $H^{1/2}( ext{T}^3)$.

Developed a linear profile decomposition for the problem.

Applied concentration-compactness and rigidity arguments to establish results.

Abstract

In this article, we investigate the global well-posedness for the defocusing, cubic nonlinear Schr\"{o}dinger equation posed on $\T^3$ with intial data lying in its critical space $H^\frac{1}{2}(\T^3)$. By establishing the linear profile decomposition, and applied this to the concentration-compactness/rigidity argument, we prove that if the solution remains bounded in the critical Sobolev space throughout the maximal lifespan, i.e. $u\in L_t^\infty{H}^\frac{1}{2}(I\times\T^3)$, then $u$ is global.