Abstract
We consider the Landau-Lifshitz flow on a bounded planar domain. An $\epsilon$ -regularity type a-priori estimate provides the analytic tool for the subsequent geometric description of the flow at isolated singularities. At forward isolated singularities where the energy is not left continuous the flow concentrates energy and develops bubbles. As in J.Qing’s bubbling-energy-equality for the harmonic map flow, the energy loss at such a singularity can be recovered as a finite sum of energies of tangent bubbles. We then clarify a known uniqueness result for the Landau-Lifshitz flow and show how non-uniqueness of extensions of the flow after point singularities is related to backward bubbling. Finally the $\epsilon$ -regularity estimate also yields a partial compactness result for sequences of smooth solutions to the Landau-Lifshitz flow with uniformly bounded energy, defined on a planar domain.
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