Christof Melcher

Existence of Partially Regular Solutions for Landau–Lifshitz Equations in ℝ3

Abstract We establish the existence of partially regular weak solutions for the Landau–Lifshitz equation in three space dimensions for smooth initial data of finite Dirichlet energy. The construction is based on Ginzburg–Landau approximation. The new key ingredient is a nonlocal representation formula for the penalty term that permits us to take advantage of the special trilinear structure of the limiting nonlinearity.

Spectral approximation of pattern-forming nonlinear evolution equations with double-well potentials of quadratic growth

This paper is concerned with the analysis of a numerical algorithm for the approximate solution of a class of nonlinear evolution problems that arise as L 2 gradient flow for the Modica-Mortola regularization of the functional v ∈ BV(T d ; {-1,1}) ↦ E(v) := γ/2 ∫ Td ∇v + 1/2 ∑ k∈ℤd σ(k)v(k) 2 . Here γ is the interfacial energy per unit length or unit area, T d is the flat torus in ℝ d , and σ is a nonnegative Fourier multiplier, that is continuous on ℝ d , symmetric in the sense that σ(ξ) = σ(-ξ) for all ξ ∈ ℝ d and that decays to zero at infinity. Such functionals feature in mathematical models of pattern-formation in micromagnetics and models of diblock copolymers. The resulting evolution equation is discretized by a Fourier spectral method with respect to the spatial variables and a modified Crank—Nicolson scheme in time. Optimal-order a priori bounds are derived on the global error in the l ∞ (0, T; L 2 (T d )) norm.

Wave-type dynamics in ferromagnetic thin films and the motion of Néel walls

We investigate the magnetization dynamics in soft ferromagnetic films with small damping. In this case, the gyrotropic nature of Landau–Lifshitz–Gilbert dynamics and the shape anisotropy effects from stray-field interactions effectively lead to a wave-type dynamics for the in-plane magnetization. We apply this result to study the motion of Neel walls in thin films and prove the existence of a travelling wave solution under a small constant forcing.

Antiskyrmions stabilized at interfaces by anisotropic Dzyaloshinskii-Moriya interactions

Chiral magnets are an emerging class of topological matter harboring localized and topologically protected vortex-like magnetic textures called skyrmions, which are currently under intense scrutiny as an entity for information storage and processing. Here, on the level of micromagnetics we rigorously show that chiral magnets can not only host skyrmions but also antiskyrmions as least energy configurations over all non-trivial homotopy classes. We derive practical criteria for their occurrence and coexistence with skyrmions that can be fulfilled by (110)-oriented interfaces depending on the electronic structure. Relating the electronic structure to an atomistic spin-lattice model by means of density functional calculations and minimizing the energy on a mesoscopic scale by applying spin-relaxation methods, we propose a double layer of Fe grown on a W(110) substrate as a practical example. We conjecture that ultra-thin magnetic films grown on semiconductor or heavy metal substrates with C2v symmetry are prototype classes of materials hosting magnetic antiskyrmions.Skyrmions, localized defects in the magnetization, can be stabilised in materials by the Dzyaloshinskii-Moriya interaction (DMI). Hoffmann et al. predict that, when the DMI is anisotropic, antiskyrmions can be formed and coexist with skyrmions, enabling studies and exploitation of their interactions.

Chiral skyrmions in the plane

Magnets without inversion symmetry are a prime example of a solid-state system featuring topological solitons on the nanoscale, and a promising candidate for novel spintronic applications. Magnetic chiral skyrmions are localized vortex-like structures, which are stabilized by antisymmetric exchange interaction, the so-called Dzyaloshinskii–Moriya interaction. In continuum theories, the corresponding energy contribution is, in contrast to the classical Skyrme mechanism from nuclear physics, of linear gradient dependence and breaks the chiral symmetry. In the simplest possible case of a ferromagnetic energy in the plane, including symmetric and antisymmetric exchange and Zeeman interaction, we show that the least energy in a class of fields with unit topological charge is attained provided the Zeeman field is sufficiently large.

Vortex motion for the Landau-Lifshitz-Gilbert equation with spin transfer torque

We study the Landau–Lifshitz–Gilbert equation for the dynamics of a magnetic vortex system. We include the spin-torque effects of an applied spin current, and we rigorously derive an equation of motion (“Thiele equation”) for vortices if the current is not too large. Our method of proof strongly utilizes the geometry of the problem in order to obtain the necessary energy estimates.

Landau--Lifshitz--Slonczewski Equations: Global Weak and Classical Solutions

We consider magnetization dynamics under the influence of a spin-polarized current, given in terms of a spin-velocity field

An Implicit Midpoint Spectral Approximation of Nonlocal Cahn--Hilliard Equations

\boldsymbol{v}

Thin-Film Limits for Landau–Lifshitz–Gilbert Equations

, governed by the following modification of the Landau--Lifshitz--Gilbert equation

Vortex Motion for the Landau-Lifshitz-Gilbert Equation with Applied Magnetic Field

\frac{\partial \boldsymbol{m}}{\partial t} + \boldsymbol{v} \cdot \nabla \boldsymbol{m}= \boldsymbol{m} \times (\alpha \, \frac{\partial \boldsymbol{m}}{\partial t} + \beta \, \boldsymbol{v} \cdot \nabla \boldsymbol{m} - \Delta \boldsymbol{m}),