Formation of singularities for equivariant 2+1 dimensional wave maps into the two-sphere

TL;DR

This paper presents numerical evidence that large energy equivariant wave maps from 2+1D Minkowski space into the two-sphere develop finite-time singularities, exhibiting a universal shrinking harmonic map behavior.

Contribution

It provides the first numerical confirmation of singularity formation and universality in equivariant wave maps into the two-sphere.

Findings

Large energy initial data lead to finite-time singularities.

Singularities exhibit universal adiabatic shrinking of harmonic maps.

Supports conjecture on universal behavior of wave map singularities.

Abstract

In this paper we report on numerical studies of the Cauchy problem for equivariant wave maps from 2+1 dimensional Minkowski spacetime into the two-sphere. Our results provide strong evidence for the conjecture that large energy initial data develop singularities in finite time and that singularity formation has the universal form of adiabatic shrinking of the degree-one harmonic map from $\mathbb{R}^2$ into $S^2$.