University of Birmingham > Talks@bham > Analysis Seminar > Long-time behaviour of solutions to the equivariant Harmonic Map Heat Flow and Landau-Lifschitz equations

Long-time behaviour of solutions to the equivariant Harmonic Map Heat Flow and Landau-Lifschitz equations

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If you have a question about this talk, please contact Yuzhao Wang.

We study the asymptotic behaviour of corotational solutions to the Harmonic Map Heat Flow from $\mathbb{R}2$ to $\mathbb{S}2.$ In particular, we give criteria in terms of the initial data: we compare their energy to the energy of the static solutions, which are setting the natural threshold between, on one hand, global existence and decay, and on the other, blow-up or lack of decay.

We first recover Struwe’s global existence result for a below threshold scenario; we also show decay to zero as time goes to infinity. The proof in this case follows the “concentration-compactness plus rigidity” approach of Kenig and Merle, originally developed for dispersive equations.

We then proceed to extend results of Gustafson, Nakanishi and Tsai for above threshold maps. Employing a characterization of blowing-up solutions and a new stability result, based on modulation arguments and spectral considerations, we show that solutions in an above threshold class (but not close to the harmonic maps) exist globally and asymptotically relax to a rescaled static solution.

If time permits, we will briefly sketch how these ideas can be used in the study of above-threshold solutions to the Landau-Lifschitz equation.

This talk is part of the Analysis Seminar series.

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