| Abstract: |
| We consider geometric evolution problems consisting of evolution equations for a closed hypersurface coupled to parabolic equations on this evolving surface. More precisely, the evolution of the hypersurface is determined by a geometric flow that depends on a quantity defined on the surface via a diffusion equation. This system arises as a gradient flow of an energy functional. Assuming suitable parabolicity conditions, we derive short-time existence for the system. The proof is based on a linearization and contraction argument.
Afterwards, several properties of the solution are analyzed. In particular, we emphasize to what extent the surface in our setting evolves in the same way as under the usual mean curvature flow. To this end, we show that the surface area is strictly decreasing but we give an example of a surface that exists for infinite times nevertheless.
Moreover, mean convexity is conserved whereas convexity is not.
We will also discuss how Willmore type flow can be coupled to the Cahn-Hilliard equation on an evolving surface. The resulting system is a coupled Cahn-Hilliard/Canham-Helfrich flow. Finally, we construct an embedded hypersurface that develops a self-intersection over time and we discuss how solutions can be computed numerically with the help of an evolving surface finite element discretization. |
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