The Logarithmic Tail of Néel Walls

A compactness result in thin-film micromagnetics and the optimality of the Néel wall

“…and therefore (137) explains the expression of the energy E 1d ε (m ) given in (11). Also observe that the chosen stray field energy is minimal because for any h : R 3 → R 3 that is 1-periodic in x 2 and satisfies (2) for ∇ · m , we have…”

“…The prediction of the logarithmic decay was formally proved by Riedel and Seeger [13]; a detailed mathematical discussion of their results was carried out by García-Cervera [8]. Finally, Melcher [11,12] rigorously established the exact logarithmic scaling for the 180 • Néel wall tails: The minimizer m 1 with m 1 (0) = 1 is symmetric around 0 (w core ∼ ε) and satisfies Figure 5). Moreover, the leading order term of the minimal energy level is min (9), (10) …”

“…We define the Néel wall as the 1-d minimizer of (11) under the boundary constraint (10). The Néel wall is a two-length-scale object: a small core (|x 1 | w core ) with fast varying rotation and a logarithmically decaying tail (w core |x 1 | 1).…”

Numerical analysis of an explicit approximation scheme for the Landau-Lifshitz-Gilbert equation

“…Guo and Hong [8] successfully carried through the argument that Struwe in [15] employed for the harmonic map heat flow to exhibit a Struwe solution in two dimensions, i.e., a partially regular solution that satisfies an energy inequality and is smooth away from a finite set of point singularities. Recently, Ko [10] in two dimensions and Melcher [12] in three dimensions independently constructed partially regular solutions to LLG smooth away from a locally finite n-dimensional parabolic Hausdorff measure set. While it is known that weak solutions are nonunique in general [2], uniqueness or nonuniqueness in the class of partially regular solutions is, however, still an open question.…”

One-dimensional Néel walls under applied external fields

“…Rigorous mathematical studies of the Néel walls are more recent and go back to the work of García-Cervera [14,16], who undertook some analysis of the associated one-dimensional variational problems and performed extensive numerical studies of the energy functional obtained by Aharoni from the full micromagnetic energy after restricting the admissible configurations to profiles which depend only on one spatial variable [21]. Melcher further studied the minimizers of the same functional in the class of magnetization configurations constrained to the film plane and established symmetry and monotonicity of the energy minimizing profiles connecting the two opposite directions of the easy axis [22]. Using a further one-dimensional thin film reduction of the micromagnetic energy introduced in [17], Capella, Melcher and Otto outlined the proof of uniqueness of the Néel wall profile and its linearized stability with respect to one-dimensional perturbations [23].…”