Maximal Solutions and Stochastic Free Boundary Formulations for Stochastic Willmore and Surface Diffusion Flows on $\R^2$

TL;DR

This paper investigates stochastic Willmore and surface diffusion flows for curves in the plane, formulating them as stochastic free boundary problems and establishing local existence and uniqueness of solutions using advanced stochastic PDE theory.

Contribution

It introduces a stochastic free boundary formulation for curve flows and applies recent quasilinear stochastic PDE theory to prove local well-posedness.

Findings

Existence and uniqueness of local strong solutions up to a maximal stopping time.

Solutions may develop singularities or shrink to a point upon blow-up.

Framework applicable to stochastic curvature-driven geometric flows.

Abstract

We study the stochastic Willmore flow and the stochastic surface diffusion flow for closed or non-closed curves on $\mathbb{R}^2$ in this paper. We equivalently formulate them as a stochastic one-phase Stefan problem (or a stochastic free boundary problem) of the curvature, which is parameterized by the arc-length, and the length of the curves. After rewriting the stochastic Stefan problem as a quasilinear parabolic evolution equation, we apply the theory for quasilinear parabolic stochastic evolution equations developed by Agresti and Veraar in 2022 to get the existence and uniqueness of a local strong solution up to a maximal stopping time that is characterized by a blow-up alternative. When the solutions blow up, the corresponding stochastic curve flows either develop singularities or shrink to a point.