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Caltech/UCLA/USC Joint Analysis Seminar

Tuesday, November 8, 2022
2:00pm to 2:50pm
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The soliton resolution conjecture for equivariant wave maps
Andrew Lawrie, Department of Mathematics, MIT,

UCLA, room MS 6221 at the Math Sciences Building

I will present a joint work with Jacek Jendrej (CRNS, Sorbonne Paris Nord) on equivariant wave maps with values in the two-sphere. We prove that every finite energy solution resolves, as time passes, into a superposition of harmonic maps (solitons) and radiation, settling the soliton resolution problem for this equation. It was proved in works of Côte, and Jia-Kenig, that such a decomposition holds along a sequence of times. We show the resolution holds continuously-in-time via a "no-return" lemma based on the virial identity. The proof combines a modulation analysis of solutions near a multi-soliton configuration with concentration compactness techniques. As a byproduct of our analysis we prove that there are no pure multi-solitons in equivariance class k=1 and no elastic collisions between pure multi-solitons in the higher equivariance classes.

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