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

Abstract:

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