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Winding for Wave Maps

June 1, 2020
Presenters: Max Engelstein
Length: 1 hour

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Wave maps are harmonic maps from a Lorentzian domain to a Riemannian target. Like solutions to many energy critical PDE, wave maps can develop singularities where the energy concentrates on arbitrary small scales but the norm stays bounded. Zooming in on these singularities yields a harmonic map (called a soliton or bubble) in the weak limit. One fundamental question is whether this weak limit is unique, that is to say, whether different bubbles may appear as the limit of different sequences of rescalings. We show by example that uniqueness may not hold if the target manifold is not analytic. Our construction is heavily inspired by Peter Topping's analogous example of a ``winding" bubble in harmonic map heat flow. However, the Hamiltonian nature of the wave maps will occasionally necessitate different arguments.

This is joint work with Dana Mendelson (U Chicago).

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