Global well-posedness and scattering for the defocusing H˙1/2-critical nonlinear Schrödinger equation in ℝ2

Abstract

We consider the Cauchy initial value problem for the defocusing quintic nonlinear Schrödinger equation in two dimensions with general data in the critical space ${\dot{H}}^{1∕2}\left({ℝ}^2\right)$. We show that if a solution remains bounded in ${\dot{H}}^{1∕2}\left({ℝ}^2\right)$ in its maximal interval of existence, then the interval is infinite and the solution scatters.