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.

Vortex Motion for the Landau–Lifshitz–Gilbert Equation with Spin-Transfer Torque

“…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.…”

The gradient flow motion of boundary vortices

“…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.…”

Existence of partially regular weak solutions to Landau–Lifshitz–Maxwell equations

“…(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.…”

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