Threshold dynamics for corotational wave maps

TL;DR

This paper investigates the behavior of corotational wave maps at the critical energy level of 8π, demonstrating global existence, scattering, or finite-time blow-up, and contrasting these dynamics with higher equivariant wave maps.

Contribution

It establishes the first example of finite-time blow-up for minimal topologically trivial wave maps at threshold energy and analyzes their global behavior.

Findings

Corotational wave maps with energy 8π can either scatter or blow up in finite time.

Constructed a finite-time blow-up solution at the threshold energy level.

Contrasts the behavior of threshold wave maps with higher equivariant cases.

Abstract

We study the dynamics of corotational wave maps from $\mathbb R^{1+2} \rightarrow \mathbb S^2$ at threshold energy. It is known that topologically trivial wave maps with energy $< 8\pi$ are global and scatter to a constant map. In this work, we prove that a corotational wave map with energy equal to $8\pi$ is globally defined and scatters in one time direction, and in the other time direction, either the map is globally defined and scatters, or the map breaks down in finite time and converges to a superposition of two harmonic maps. The latter behavior stands in stark contrast to higher equivariant wave maps with threshold energy which have been proven to be globally defined for all time. Using techniques developed in this paper, we also construct a corotational wave map with energy $= 8\pi$ which blows up in finite time. The blow-up solution we construct provides the first example of a minimal topologically trivial non-dispersing solution to the full wave map evolution.