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

Qi Yan

TL;DR

The paper develops a stochastic-geometric framework for planar Willmore and curve-diffusion flows by reformulating them as stochastic one-phase Stefan problems for curvature $k$ and length $L$ under arc-length parameterization. It casts the resulting evolution into a quasilinear stochastic evolution equation on suitable UMD-Banach spaces and proves the existence and uniqueness of $L^p$-maximal local solutions using the $H^{ abla}$-calculus and $R$-boundedness machinery, valid for both 1D and infinite-dimensional Brownian noise and for closed and non-closed curves. A rigorous blow-up criterion shows that the flow fails only when either the curvature becomes unbounded or the curve shrinks to a point. The work provides a robust pathway to analyze stochastic curvature-driven flows via free-boundary problems and advanced stochastic-analytic tools, with potential extensions to more general curve-dynamics and Riemannian settings.

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.

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

TL;DR

The paper develops a stochastic-geometric framework for planar Willmore and curve-diffusion flows by reformulating them as stochastic one-phase Stefan problems for curvature and length under arc-length parameterization. It casts the resulting evolution into a quasilinear stochastic evolution equation on suitable UMD-Banach spaces and proves the existence and uniqueness of -maximal local solutions using the -calculus and -boundedness machinery, valid for both 1D and infinite-dimensional Brownian noise and for closed and non-closed curves. A rigorous blow-up criterion shows that the flow fails only when either the curvature becomes unbounded or the curve shrinks to a point. The work provides a robust pathway to analyze stochastic curvature-driven flows via free-boundary problems and advanced stochastic-analytic tools, with potential extensions to more general curve-dynamics and Riemannian settings.

Abstract

We study the stochastic Willmore flow and the stochastic surface diffusion flow for closed or non-closed curves on 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.

Paper Structure

This paper contains 11 sections, 9 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stochastic integration
  4. The space $\gamma(\mathcal{H},X)$
  5. The stochastic integral
  6. $R$-boundedness and $H^{\infty}$-calculus
  7. Equivalence of a General curve flow and a one-phase Stefan problem
  8. Stochastic Willmore flow with one-dimensional Brownian motion
  9. Stochastic Willmore flow with infinite-dimensional Brownian motion
  10. Stochastic curve diffusion flow
  11. Acknowledgement

Key Result

Theorem 2.1

Let $X$ be a UMD Banach space, and $G$ be an $\mathcal{F}$-adapted step process in $\gamma(H;X)$. Then for all $p\in (0,\infty)$, one has the two-sided estimate with implicit constants depending only on $p$ and the UMD constant of $X$. In particular, the stochastic integral can be continued to a linear and bounded operator

Theorems & Definitions (13)

  • Theorem 2.1
  • Lemma 4.1
  • proof : Proof of Lemma \ref{['lemma']}
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • Theorem 4.4
  • Theorem 4.5
  • Remark 4.6
  • ...and 3 more