Abels, Helmut and Buerger, Felicitas and Garcke, Harald (2023) Qualitative properties for a system coupling scaled mean curvature flow and diffusion. JOURNAL OF DIFFERENTIAL EQUATIONS, 349. pp. 236-268. ISSN 0022-0396, 1090-2732
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We consider a system consisting of a geometric evolution equation for a hypersurface and a parabolic equation on this evolving hypersurface. More precisely, we discuss mean curvature flow scaled with a term that depends on a quantity defined on the surface coupled to a diffusion equation for that quantity. Several properties of solutions are analyzed. Emphasis is placed on to what extent the surface in our setting qualitatively evolves similar as for the usual mean curvature flow. To this end, we show that the surface area is strictly decreasing but gives an example of a surface that exists for infinite times nevertheless. Moreover, mean convexity is conserved whereas convexity is not. Finally, we construct an embedded hypersurface that develops a self-intersection in the course of time. Additionally, a formal explanation of how our equations can be interpreted as a gradient flow is included.(c) 2022 Elsevier Inc. All rights reserved.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | ; Mean curvature flow; Diffusion equation on surfaces; Geometric evolution equation; Convexity; Maximum principle |
| Subjects: | 500 Science > 510 Mathematics |
| Divisions: | Mathematics Mathematics > Prof. Dr. Helmut Abels |
| Depositing User: | Dr. Gernot Deinzer |
| Date Deposited: | 30 Jan 2024 15:38 |
| Last Modified: | 30 Jan 2024 15:38 |
| URI: | https://pred.uni-regensburg.de/id/eprint/59233 |
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