Stability of Self-similar Solutions to Geometric Flows

TL;DR

This paper demonstrates the stability of self-similar solutions to various geometric flows, such as mean curvature, surface diffusion, and Willmore flow, under small initial perturbations, showing they remain asymptotically self-similar over time.

Contribution

It provides a unified framework for proving stability of self-similar solutions across multiple geometric flows using linearized operator estimates and compactness arguments.

Findings

Self-similar solutions are stable under small Lipschitz perturbations.

Perturbed solutions tend to become asymptotically self-similar.

The approach unifies stability analysis for different flows.

Abstract

We show that self-similar solutions for the mean curvature flow, surface diffusion and Willmore flow of entire graphs are stable upon perturbations of initial data with small Lipschitz norm. Roughly speaking, the perturbed solutions are asymptotically self-similar as time tends to infinity. Our results are built upon the global analytic solutions constructed by Koch and Lamm \cite{KochLamm}, the compactness arguments adapted by Asai and Giga \cite{Giga2014}, and the spatial equi-decay properties on certain weighted function spaces. The proof for all of the above flows are achieved in a unified framework by utilizing the estimates of the linearized operator.