arXiv subjects
Publications and source records attributed to Christof Melcher.
We establish global existence, uniqueness, regularity and long-time asymptotics of strong solutions to the half-harmonic heat flow and dissipative Landau-Lifshitz equation, valid for initial data that is small in the homogeneous Sobolev norm $\dot{H}^{\frac n 2}$ for space dimensions $n \le 3$.
We show that chiral symmetry breaking enables traveling domain wall solution for the conservative Landau-Lifshitz equation of a uniaxial ferromagnet with Dzyaloshinskii-Moriya interaction. In contrast to related domain wall models including stray-field based anisotropy, traveling wave solutions are not found in closed form. For the construction we follow a topological approach and provide details of solutions by means of numerical calculations.
We examine topological solitons in a minimal variational model for a chiral magnet, so-called chiral skyrmions. In the regime of large background fields, we prove linear stability of axisymmetric chiral skyrmions under arbitrary perturbations in the energy space, a long-standing open question in physics literature. Moreover, we show strict local minimality of axisymmetric chiral skyrmions and nearby existence of moving soliton solution for the Landau-Lifshitz-Gilbert equation driven by a small spin transfer torque.
We examine a stochastic Landau-Lifshitz-Gilbert equation based on an exchange energy functional containing second-order derivatives of the unknown field. Such regularizations are featured in advanced micromagnetic models recently introduced in connection with nanoscale topological solitons. We show that, in contrast to the classical stochastic Landau-Lifshitz-Gilbert equation based on the Dirichlet energy alone, the regularized equation is solvable in the stochastically strong sense. As a consequence it preserves the topology of the initial data, almost surely.
Chiral magnets are an emerging class of topological matter harbouring localized and topologically protected vortex-like magnetic textures called skyrmions, which are currently under intense scrutiny as a new entity for information storage and processing. Here, on the level of micromagnetics we rigorously show that chiral magnets cannot 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 in dependence 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 applying spin-relaxation methods, we propose a double layer of Fe grown on a W(110) substrate as a practical example. We conjecture that ultrathin magnetic films grown on semiconductor or heavy metal substrates with $C_{2v}$ symmetry are prototype classes of materials hosting magnetic antiskyrmions.
The Landau-Lifshitz-Gilbert-Slonczewski equation describes magnetization dynamics in the presence of an applied field and a spin polarized current. In the case of axial symmetry and with focus on one space dimension, we investigate the emergence of space-time patterns in the form of wavetrains and coherent structures, whose local wavenumber varies in space. A major part of this study concerns existence and stability of wavetrains and of front- and domain wall-type coherent structures whose profiles asymptote to wavetrains or the constant up-/down-magnetizations. For certain polarization the Slonczewski term can be removed which allows for a more complete charaterization, including soliton-type solutions. Decisive for the solution structure is the polarization parameter as well as size of anisotropy compared with the difference of field intensity and current intensity normalized by the damping.
We consider the variant of a stochastic parabolic Ginzburg-Landau equation that allows for the formation of point defects of the solution. The noise in the equation is multiplicative of the gradient type. We show that the family of the Jacobians associated to the solution is tight on a suitable space of measures. Our main result is the characterization of the limit points of this family. They are concentrated on finite sums of delta measures with integer weights. The singular set of the solution coincides with the points at which the delta measures are centered.
We examine lower order perturbations of the harmonic map prob- lem from $\mathbb{R}^2$ to $\mathbb{S}^2$ including chiral interaction in form of a helicity term that prefers modulation, and a potential term that enables decay to a uniform background state. Energy functionals of this type arise in the context of magnetic systems without inversion symmetry. In the almost conformal regime, where these perturbations are weighted with a small parameter, we examine the existence of relative minimizers in a non-trivial homotopy class, so-called chiral skyrmions, strong compactness of almost minimizers, and their asymptotic limit. Finally we examine dynamic stability and compactness of almost minimizers in the context of the Landau-Lifshitz-Gilbert equation including spin-transfer torques arising from the interaction with an external current.
We study the Landau-Lifshitz-Gilbert equation for the dynamics of a magnetic vortex system. We present a PDE-based method for proving vortex dynamics that does not rely on strong well-preparedness of the initial data and allows for instantaneous changes in the strength of the gyrovector force due to bubbling events. The main tools are estimates of the Hodge decomposition of the supercurrent and an analysis of the defect measure of weak convergence of the stress energy tensor. Ginzburg-Landau equations with mixed dynamics in the presence of excess energy are also discussed.
We prove existence, uniqueness and asymptotics of global smooth solutions for the Landau-Lifshitz-Gilbert equation in dimension $n \ge 3$, valid under a smallness condition of initial gradients in the $L^n$ norm. The argument is based on the method of moving frames that produces a covariant complex Ginzburg-Landau equation, and a priori estimates that we obtain by the method of weighted-in-time norms as introduced by Fujita and Kato.