2007 # The gradient flow motion of boundary vortices

**Abstract:** We consider the gradient flow of a family of energy functionals describing the formation of boundary vortices in thin magnetic films. We obtain motion laws for the singularities in all time scalings by using the method of Γ -convergence of gradient flows.
RésuméOn considère le flot de gradient d'une famille de fonctionnelles d'énergie qui décrivent la formation de vortex dans les films magnétiques minces. Ces singularités se forment à la frontière, et nous obtenons leurs équations du mouvement, pour tous les s…

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“…Formulas (19) and 22, on the other hand, are not proved in [21], and (20) needs another derivation for equation (10). Note, however, that we do obtain the statement a ∈ H 1 (0, T ; Ω d ) from the previous work.…”

confidence: 70%

“…Formulas (19) and 22, on the other hand, are not proved in [21], and (20) needs another derivation for equation (10). Note, however, that we do obtain the statement a ∈ H 1 (0, T ; Ω d ) from the previous work.…”

confidence: 70%

“…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%

“…In one dimension, the perimeter functional has no interesting dynamics, so the behaviour of the evolution equation (without obstacles) should be governed by the next order Γ-limit. At a simple step function χ [r1,r2] on the real line we can modify arguments from [Kur06,Kur07] for closely related energies to see that…”

confidence: 99%

“…Here we follow the ideas in [33] (see also [5,21,22]) where a method to prove the convergence of gradient flows for Γ-converging energy functionals was presented and applied to derive the limiting dynamics of vortices for the heat flow of the Ginzburg-Landau energy. However, we emphasize that the convergence of the gradient flow cannot follow from the Γ-convergence of the energy functionals only and extra conditions are required for the interplay of the convergence of the energy and the dissipation potentials.…”

confidence: 99%