Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case

Abstract

We establish global wellposedness and scattering for the H 1 H^{1} - critical defocusing NLS in 3D i u t + Δ u − u | u | 4 = 0 \begin{equation*}iu_{t}+\Delta u - u|u|^{4}=0 \end{equation*} assuming radial data ϕ ∈ H s \phi \in H^{s} , s ≥ 1 s\geq 1 . In particular, it proves global existence of classical solutions in the radial case. The same result is obtained in 4D for the equation i u t + Δ u − u | u | 2 = 0. \begin{equation*}iu_{t}+\Delta u -u|u|^{2} =0. \end{equation*}