Scattering of Wave Maps from ${\mathbb{R}}^{2+1}$ to general targets

Abstract

We show that smooth, radially symmetric wave maps $U$ from ${\mathbb{R}}^{2+1}$ to a compact target manifold $N$, where ${\partial }_{r}U$ and ${\partial }_{t}U$ have compact support for any fixed time, scatter. The result will follow from the work of Christodoulou and Tahvildar-Zadeh, and Struwe, upon proving that for ${\lambda }^{\prime }\in (0,1)$, energy does not concentrate in the set $$K_{5/8T,7/8T}^{{\lambda }^{\prime }}=(x,t)\in {\mathbb{R}}^{2+1}|5pt|x|\le {\lambda }^{\prime }t,t\in [(5/8)T,(7/8)T].$$