Uniqueness of Two‐Bubble Wave Maps in High Equivariance Classes

Abstract

This is the second part of a two-paper series that establishes the uniqueness and regularity of a threshold energy wave map that does not scatter in both time directions.Consider the -valued equivariant energy critical wave maps equation on , with equivariance class . It is known that every topologically trivial wave map with energy less than twice that of the unique -equivariant harmonic map scatters in both time directions. We study maps with precisely the threshold energy E E . In the first part of the series [15] we gave a refined construction of a threshold wave map that asymptotically decouples into a superposition of two harmonic maps (bubbles), one of which is concentrating in scale. In this paper, we show that this solution is the unique (up to the natural invariances of the equation) two-bubble wave map. Combined with our earlier work [14] we obtain an exact description of every threshold wave map.

Soliton resolution for the energy-critical nonlinear wave equation in the radial case

“…Viewed in forward time, this means that the 2-soliton structure emerges from pure radiation, and constitutes an orbit connecting two different dynamical behaviors. We later showed in [37,38] that u (2) (t) is the unique 2-bubble solution up to sign, translation, and scaling in equivariance classes k ≥ 4. While we do not consider such refined two-directional analysis here, a relatively straightforward corollary of the proof of Theorem 1 is that there can be no elastic collisions of pure multi-bubbles, which we formulate as a proposition below.…”