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

Analysis of PDEs