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Register a new userVariational principles of micromagnetics revisited
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We revisit the basic variational formulation of the minimization problem associated with the micromagnetic energy, with an emphasis on the treatment of the stray field contribution to the energy, which is intrinsically non-local. Under minimal assumptions, we establish three distinct variational principles for the stray field energy: a minimax principle involving magnetic scalar potential and two minimization principles involving magnetic vector potential. We then apply our formulations to the dimension reduction problem for thin ferromagnetic shells of arbitrary shapes.
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In this paper, we introduce a new framework for parametrization schemes (PS) in GFD. Using the theory of controlled rough paths, we derive a class of rough geophysical fluid dynamics (RGFD) models as critical points of rough action functionals. These RGFD models characterize Lagrangian trajectories in fluid dynamics as geometric rough paths (GRP) on the manifold of diffeomorphic maps. Three constrained variational approaches are formulated for the derivation of these models. The first is the Clebsch formulation, in which the constraints are imposed as rough advection laws. The second is the Hamilton-Pontryagin formulation, in which the constraints are imposed as right-invariant rough vector fields. The third is the Euler--Poincare formulation in which the variations are constrained. These variational principles lead directly to the Lie--Poisson Hamiltonian formulation of fluid dynamics on geometric rough paths. The GRP framework preserves the geometric structure of fluid dynamics obtained by using Lie group reduction to pass from Lagrangian to Eulerian variational principles, thereby yielding a rough formulation of the Kelvin circulation theorem. The rough-path variational approach includes non-Markovian perturbations of the Lagrangian fluid trajectories. In particular, memory effects can be introduced through this formulation through a judicious choice of the rough path (e.g. a realization of a fractional Brownian motion). In the special case when the rough path is a realization of a semimartingale, we recover the SGFD models in Holm (2015). However, by eliminating the need for stochastic variational tools, we retain a pathwise interpretation of the Lagrangian trajectories. In contrast, the Lagrangian trajectories in the stochastic framework are described by stochastic integrals which do not have a pathwise interpretation. Thus, the rough path formulation restores this property.
The determination of the speed of travelling fronts of the scalar reaction diffusion equation has been the subject of much study. Using different approaches seemingly disconnected variational principles have been established. The purpose of this work is to show the connection between them. For monostable reaction terms, we prove that a principle established by Hadeler and Rothe in 1975 and a second one by Benguria and Depassier in 1996 are logically equivalent, that is, either can be derived from the other. Two variational principles, formulated for arbitrary reaction terms, are shown to be related by a suitable change of variables. Finally a variational principle proven for monostable reaction terms is shown to be a formulation of the two previous ones in yet another independent variable.
In this paper, we study the thin-film limit of the micromagnetic energy functional in the presence of bulk Dzyaloshinskii-Moriya interaction (DMI). Our analysis includes both a stationary $Gamma$-convergence result for the micromagnetic energy, as well as the identification of the asymptotic behavior of the associated Landau-Lifshitz-Gilbert equation. In particular, we prove that, in the limiting model, part of the DMI term behaves like the projection of the magnetic moment onto the normal to the film, contributing this way to an increase in the shape anisotropy arising from the magnetostatic self-energy. Finally, we discuss a convergent finite element approach for the approximation of the time-dependent case and use it to numerically compare the original three-dimensional model with the two-dimensional thin-film limit.
We study a geometric variational problem for sets in the plane in which the perimeter and a regularized dipolar interaction compete under a mass constraint. In contrast to previously studied nonlocal isoperimetric problems, here the nonlocal term asymptotically localizes and contributes to the perimeter term to leading order. We establish existence of generalized minimizers for all values of the dipolar strength, mass and regularization cutoff and give conditions for existence of classical minimizers. For subcritical dipolar strengths we prove that the limiting functional is a renormalized perimeter and that for small cutoff lengths all mass-constrained minimizers are disks. For critical dipolar strength, we identify the next-order $Gamma$-limit when sending the cutoff length to zero and prove that with a slight modification of the dipolar kernel there exist masses for which classical minimizers are not disks.
We present a characterization of the domain wall solutions arising as minimizers of an energy functional obtained in a suitable asymptotic regime of micromagnetics for infinitely long thin film ferromagnetic strips in which the magnetization is forced to lie in the film plane. For the considered energy, we provide existence, uniqueness, monotonicity, and symmetry of the magnetization profiles in the form of 180$^circ$ and 360$^circ$ walls. We also demonstrate how this energy arises as a $Gamma$-limit of the reduced two-dimensional thin film micromagnetic energy that captures the non-local effects associated with the stray field, and characterize its respective energy minimizers.
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