Two-bubble dynamics for threshold solutions to the wave maps equation

TL;DR

This paper analyzes the behavior of solutions to the energy-critical wave maps equation at the threshold energy level, revealing conditions under which solutions scatter or converge to a superposition of harmonic maps, using advanced analytical techniques.

Contribution

It establishes the dynamics of threshold solutions for wave maps with two-bubble configurations, combining concentration-compactness and modulation analysis.

Findings

Solutions at threshold energy either scatter or form a superposition of harmonic maps.

The asymptotic scales of the harmonic maps are characterized.

The analysis extends understanding of energy-critical wave map dynamics.

Abstract

We consider the energy-critical wave maps equation $\mathbb R^{1+2} \to \mathbb S^2$ in the equivariant case, with equivariance degree $k \geq 2$. It is known that initial data of energy $ < 8k\pi$ and topological degree zero leads to global solutions that scatter in both time directions. We consider the threshold case of energy $8k\pi$. We prove that the solution is defined for all time and either scatters in both time directions, or converges to a superposition of two harmonic maps in one time direction and scatters in the other time direction. In the latter case, we describe the asymptotic behavior of the scales of the two harmonic maps. The proof combines the classical concentration-compactness techniques of Kenig-Merle with a modulation analysis of interactions of two harmonic maps in the absence of excess radiation.