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

TL;DR

This paper proves that smooth, radially symmetric wave maps from 2+1 dimensional space-time to compact targets with compactly supported derivatives scatter, extending previous work by establishing non-concentration of energy in specific regions.

Contribution

It demonstrates scattering for a class of wave maps by proving energy does not concentrate in certain spacetime regions, building on prior foundational results.

Findings

Wave maps with compactly supported derivatives scatter.

Energy does not concentrate in specified spacetime regions.

Extension of previous scattering results 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 $\lambda' \in (0,1)$, energy does not concentrate in the set $$K_{5/8T,7/8T}^{\lambda'} = {(x,t) \in \mathbb R^{2+1} | {5pt} |x| \leq \lambda' t, t \in [(5/8)T,(7/8)T]}.$$