The nonlocal mean curvature flow of periodic graphs

TL;DR

This paper proves the well-posedness and exponential convergence of solutions to the nonlocal mean curvature flow of periodic graphs in certain function spaces, using a reformulation as a parabolic evolution problem.

Contribution

It establishes the well-posedness and long-term behavior of nonlocal mean curvature flow for periodic graphs in subcritical H"older spaces, extending previous results to a nonlocal setting.

Findings

Solutions exist uniquely in specified function spaces.

Solutions close to their mean converge exponentially.

The flow is shown to be of parabolic type via localization.

Abstract

We establish the well-posedness of the nonlocal mean curvature flow of order ${\alpha\in(0,1)}$ for periodic graphs on $\mathbb{R}^n$ in all subcritical little H\"older spaces ${\rm h}^{1+\beta}(\mathbb{T}^n)$ with $\beta\in(0,1)$. Furthermore, we prove that if the solution is initially sufficiently close to its integral mean in ${\rm h}^{1+\beta}(\mathbb{T}^n)$, then it exists globally in time and converges exponentially fast towards a constant. The proofs rely on the reformulation of the equation as a quasilinear evolution problem, which is shown to be of parabolic type by a direct localization approach, and on abstract parabolic theories for such problems.