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

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