An asymptotic expansion of two-bubble wave maps in high equivariance classes

Abstract

This is the first 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 ${\mathbb{𝕊}}^2$-valued equivariant energy critical wave maps equation on ${ℝ}^{1+2}$, with equivariance class $k\ge 4$. It is known that every topologically trivial wave map with energy less than twice that of the unique $k$-equivariant harmonic map $Q_k$ scatters in both time directions. We study maps with precisely the threshold energy $\mathcal{ℰ}=2\mathcal{ℰ}(Q_k)$.

In this paper, we give a refined construction of a wave map with threshold energy that converges to a superposition of two harmonic maps (bubbles), asymptotically decoupling in scale. We show that this two-bubble solution possesses $H^2$ regularity. We give a precise dynamical description of the modulation parameters as well as an expansion of the map into profiles. In the next paper in the series, we show that this solution is unique (up to the natural invariances of the equation) relying crucially on the detailed properties of the solution constructed here. Combined with our earlier work (in 2018), we can now give an exact description of every threshold wave map.