#### Vol. 15, No. 2, 2022

 Download this article For screen For printing
 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print) Author Index To Appear Other MSP Journals
An asymptotic expansion of two-bubble wave maps in high equivariance classes

### Jacek Jendrej and Andrew Lawrie

Vol. 15 (2022), No. 2, 327–403
##### 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 $\mathsc{ℰ}=2\mathsc{ℰ}\left({Q}_{k}\right)$.

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.

##### Keywords
wave maps, multisolitons
##### Mathematical Subject Classification 2010
Primary: 35C08, 35C20, 35L05, 35Q99, 37K40
##### Milestones
Received: 23 March 2020
Accepted: 27 October 2020
Published: 12 April 2022
##### Authors
 Jacek Jendrej CNRS and LAGA, Université Sorbonne Paris Nord Villetaneuse France Andrew Lawrie Department of Mathematics Massachusetts Institute of Technology Cambridge, MA United States