Continuous time soliton resolution for two-bubble equivariant wave maps

Abstract

We consider the energy-critical wave maps equation from 1+2 dimensional Minkowski space into the 2-sphere, 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. If the equivariance degree equals one or two, we deduce, as a consequence of sequential soliton resolution results of C\^ote, and Jia and Kenig, 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.