Short Time Existence for Coupling of Scaled Mean Curvature Flow and Diffusion

TL;DR

This paper establishes a short-term existence result for a coupled system involving scaled mean curvature flow and a diffusion equation on an evolving hypersurface, using a splitting and contraction approach.

Contribution

It introduces a novel method to prove short-time existence for a coupled geometric and parabolic system on immersed hypersurfaces.

Findings

Short time existence is guaranteed for the coupled system.

The method provides a uniform lower bound on the existence time.

Applicable to small variations in initial height functions.

Abstract

We prove a short time existence result for a system consisting of a geometric evolution equation for a hypersurface and a parabolic equation on this evolving hypersurface. More precisely, we discuss a mean curvature flow scaled with a term that depends on a quantity defined on the surface coupled to a diffusion equation for that quantity. The proof is based on a splitting ansatz, solving both equations separately using linearization and a contraction argument. Our result is formulated for the case of immersed hypersurfaces and yields a uniform lower bound on the existence time that allows for small changes in the initial value of the height function.