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%0 Thesis
%A Meinert, Daniel
%T Heat flow methods and saddle configurations for spherical magnets
%I RWTH Aachen University
%V Dissertation
%C Aachen
%M RWTH-2025-09948
%P 1 Online-Ressource : Illustrationen
%D 2025
%Z Veröffentlicht auf dem Publikationsserver der RWTH Aachen University
%Z Dissertation, RWTH Aachen University, 2025
%X 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.
%F PUB:(DE-HGF)11
%9 Dissertation / PhD Thesis
%R 10.18154/RWTH-2025-09948
%U https://publications.rwth-aachen.de/record/1022320