Continuous time soliton resolution for two-bubble equivariant wave maps

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.

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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