Mathematical Research Letters

Volume 29 (2022)

Number 6

Continuous time soliton resolution for two-bubble equivariant wave maps

Pages: 1745 – 1766



Jacek Jendrej (CNRS and LAGA, Université Sorbonne Paris Nord, Villetaneuse, France)

Andrew Lawrie (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.)


We consider the energy-critical wave maps equation $\mathbb{R}^{1+2} \to \mathbb{S}^2$ in the equivariant case. We prove that if a wave map decomposes, along a sequence of times, into a superposition of at most two rescaled harmonic maps (bubbles) and radiation, then such a decomposition holds for continuous time. We deduce, as a consequence of sequential soliton resolution results of Côte [5], and Jia and Kenig [25], that any topologically trivial equivariant wave map with energy less than four times the energy of the bubble asymptotically decomposes into (at most two) bubbles and radiation.

J. Jendrej was supported by ANR-18-CE40-0028 project ESSED. A. Lawrie was supported by NSF grant DMS-1954455, a Sloan Research Fellowship, and the Solomon Buchsbaum Research Fund.

Received 8 March 2021

Accepted 1 June 2021

Published 4 May 2023