- AutorIn
- Tim Bastian Laux
- Titel
- Convergence of phase-field models and thresholding schemes via the gradient flow structure of multi-phase mean-curvature flow
- Zitierfähige Url:
- https://nbn-resolving.org/urn:nbn:de:bsz:15-qucosa2-158723
- Datum der Einreichung
- 26.10.2016
- Datum der Verteidigung
- 12.04.2017
- Abstract (EN)
- This thesis is devoted to the rigorous study of approximations for (multi-phase) mean curvature flow and related equations. We establish convergence towards weak solutions of the according geometric evolution equations in the BV-setting of finite perimeter sets. Our proofs are of variational nature in the sense that we use the gradient flow structure of (multi-phase) mean curvature flow. We study two classes of schemes, namely phase-field models and thresholding schemes. The starting point of our investigation is the fact that both, the Allen-Cahn Equation and the thresholding scheme, preserve this gradient flow structure. The Allen-Cahn Equation is a gradient flow itself, while the thresholding scheme is a minimizing movements scheme for an energy that Γ-converges to the total interfacial energy. In both cases we can incorporate external forces or a volume-constraint. In the spirit of the work of Luckhaus and Sturzenhecker (Calc. Var. Partial Differential Equations 3(2):253–271, 1995), our results are conditional in the sense that we assume the time-integrated energies to converge to those of the limit. Although this assumption is natural, it is not guaranteed by the a priori estimates at hand.
- Freie Schlagwörter (EN)
- mean curvature flow, thresholding, Allen-Cahn Equation, gradient flow
- Klassifikation (DDC)
- 500
- Den akademischen Grad verleihende / prüfende Institution
- Universität Leipzig, Leipzig
- URN Qucosa
- urn:nbn:de:bsz:15-qucosa2-158723
- Veröffentlichungsdatum Qucosa
- 13.07.2017
- Dokumenttyp
- Dissertation
- Sprache des Dokumentes
- Englisch