% IMPORTANT: The following is UTF-8 encoded. This means that in the presence
% of non-ASCII characters, it will not work with BibTeX 0.99 or older.
% Instead, you should use an up-to-date BibTeX implementation like “bibtex8” or
% “biber”.
@PHDTHESIS{Meinert:1022320,
author = {Meinert, Daniel},
othercontributors = {Melcher, Christof and Westdickenberg, Maria Gabrielle and
Gustafson, Stephen},
title = {{H}eat flow methods and saddle configurations for spherical
magnets},
school = {RWTH Aachen University},
type = {Dissertation},
address = {Aachen},
publisher = {RWTH Aachen University},
reportid = {RWTH-2025-09948},
pages = {1 Online-Ressource : Illustrationen},
year = {2025},
note = {Veröffentlicht auf dem Publikationsserver der RWTH Aachen
University; Dissertation, RWTH Aachen University, 2025},
abstract = {We investigate a geometric model of a micromagnetic system
on a curved surface described by maps between 2-spheres.
This system is modeled by an energy functional that consists
of exchange energy and easy-normal anisotropy. Critical
points of this energy functional are called skyrmions and
exist because of the stabilizing effect of the curvature.
Despite its apparent simplicity, this toy model already
exhibits interesting features. The main goal of this thesis
is to prove the existence of skyrmions that are saddle
points of the energy functional. The existence of minimizers
with additional conditions on mapping degree and/or
symmetries was shown in previous works. Since the
corresponding Euler–Lagrange equation is semi linear in
nature, it is difficult to find solutions other than the
ones obtained through the direct method in the calculus of
variations. To overcome this difficulty, we use a heat flow
method by transforming the elliptic equation into a
parabolic one. This is inspired by the work of Eells and
Sampson who used this approach in the study of harmonic
maps, to which our problem is closely related. In a first
result, we prove the existence of a weak solution to the
heat flow equation away from finitely many points in
space-time at which the solution blows up. This is a result
analogous to that of Struwe for the harmonic map heat flow.
Our proof closely follows Struwe’s as the additional terms
from the anisotropy are of lower order. The long time limit
of solutions to the heat flow equation is a critical points
of the energy functional. In a second result, we show that
for a special class of axisymmetric maps no blowup can
occur, meaning that full regularity is retained in the limit
of time going to infinity. Then in a third step, we show
that the skyrmion solutions obtained in this way are saddle
points of the energy functional in a specific parameter
range of the model. This is done by showing that the
critical point is neither a local minimizer nor a local
maximizer of the energy functional. The proof is based on
the observation that these maps possess an additional
symmetry, which we then break manually by a small
perturbation to construct a map with lower energy. Finally,
we carry out numerical simulations of the heat flow equation
in the axisymmetric setting to visualize the skyrmion
solutions and to obtain a better understanding of their
properties.},
cin = {113110 / 110000},
ddc = {510},
cid = {$I:(DE-82)113110_20140620$ / $I:(DE-82)110000_20140620$},
pnm = {GRK 2326 - GRK 2326: Energie, Entropie und Dissipative
Dynamik (320021702)},
pid = {G:(GEPRIS)320021702},
typ = {PUB:(DE-HGF)11},
doi = {10.18154/RWTH-2025-09948},
url = {https://publications.rwth-aachen.de/record/1022320},
}
guest ::