Mathematical aspects of a dynamic theory of deformable ferromagnets
Recently De Simone and Podio Guidugli proposed a model for the description of the dynamics of deformable ferromagnets in the presence of microstructures. Bertsch, Podio Guidugli and Valente have started the mathematical analysis of the so-called Landau-Lifschitz-Gilbert equation, which is satisfied by the magnetization in the special case of soft ferromagnets at rest, showing that several approximation procedures can be used to construct solutions. An additional simplification of the equation leads to the dynamic harmonic-map equation. Recently it was shown by Bertsch, Dal Passo and Van der Hout that the initial-boundary value problem for the latter equation has more than one solution. We conjecture that this nonuniqueness phenomenon continues to hold for the Landau-Lifschitz-Gilbert equation, and the question arises which is the physically relevant solution and how to construct it. The answer to these questions are not yet known, but a preliminary result by Dal Passo and Vilucchi suggests that different approximation methods can lead to different solutions. One of them would correspond to the physical phenomenon of instantaneous dissipation of magnetic energy, while the second one would model the spatial concentration of a finite amount of energy on a set of measure zero.