2005 # Moving boundary vortices for a thin-film limit in micromagnetics

**Abstract:** We study the limiting behavior of solutions of the Landau-Lifshitz-Gilbert equation belonging to thin films of ferromagnetic materials. In the appropriate time scale and under reasonable conditions, there is a subsequence converging to a map that has vortices at two boundary points. The vortices move Hölder-continuously in time, and the map satisfies a formal Euler-Lagrange equation away from the vortices.

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“…Finally, for other mathematical works studying the motion of singularities in ferromagnets, we mention [6,27] for the motion of Néel walls and [19,29] for boundary vortices.…”

confidence: 99%

“…Finally, for other mathematical works studying the motion of singularities in ferromagnets, we mention [6,27] for the motion of Néel walls and [19,29] for boundary vortices.…”

confidence: 99%

“…Other rescalings lead to trivial motion laws for the boundary vortices. Related results for a different model for boundary vortices were found by Moser [22], but without the exact motion law. Moser actually studies the Landau-Lifshitz-Gilbert (LLG) equations instead of the gradient flow, which is the correct model for the evolution of magnetic structures.…”

confidence: 85%

“…(1.8) coupling with static Maxwell equations (1.2) are about the stability of static solution which matches u(−∞) = −e 1 and u(+∞) = e 1 [7], the result on the evolution of boundary vortices [33], and boundary layers [6] in some special limit regimes. Other dynamic behaviors of the domains and domain walls can be found in [22,28] and references therein.…”

confidence: 99%

“…In variety of multiscale regimes, the authors of [3,[11][12][13]19,[23][24][25][31][32][33] investigated the domain structures.…”

confidence: 99%