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Zbl 1158.35098
Moser, Roger
Energy concentration for the Landau-Lifshitz equation. (English)
[J] Ann. Inst. Henri Poincaré, Anal. Non Linéaire 25, No. 5, 987-1013 (2008). ISSN 0294-1449

Given a weakly convergent sequence of solutions of the Landau-Lifshitz equation $\frac{\partial u}{\partial t} + a u \wedge (u \wedge \Delta u) + bu \wedge \Delta u =0$ in a three-dimensional domain $\Omega$, the energy density may concentrate in such a limit on a certain subset of $\Omega$. The author shows that almost for all values of $t$ such a subset may be interpreted as a collection of generalized curves in $\Omega$ and the evolution of these curves in $t$ is partially described by a certain generalization of the mean curvature flow which is $\frac{\partial F}{\partial t} = a H_t - b\tau_t \wedge H_t$, where $F:S^1 \times [0,T) \to \Omega$ is a family of curves depending on $t$, $H_t$ is the curvature vector field, and $\tau_t$ is the unit tangent vector field.
[Iskander A. Taimanov (Novosibirsk)]

MSC 2000:: *35Q60 PDE of electromagnetic theory and optics
53C44 Geometric evolution equations (mean curvature flow)

Keywords: Landau-Lifshitz equation; energy concentration; mean curvature flow

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