Global well-posedness for the defocusing cubic nonlinear Schrödinger equation on ${\mathbb{T}}^3$

Abstract

In this article, we investigate the global well-posedness for the defocusing, cubic nonlinear Schrödinger 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.

arXiv admin note: text overlap with arXiv:2011.12925 by other authors