Short time existence and smoothness of the nonlocal mean curvature flow of graphs

TL;DR

This paper proves short-term existence, uniqueness, and smoothness of solutions for graphs evolving under fractional mean curvature flow, using an analytic semigroup approach to establish regularity and smoothness over time.

Contribution

It introduces a novel analytic semigroup method to demonstrate short-time existence and smoothness of nonlocal mean curvature flow for graphs, with optimal regularity results.

Findings

Established short-time existence and uniqueness of solutions.

Proved solutions become infinitely smooth for positive times.

Provided regularity estimates depending on initial graph smoothness.

Abstract

We consider the geometric evolution problem of entire graphs moving by fractional mean curvature. For this, we study the associated nonlocal quasilinear evolution equation satisfied by the family of graph functions. We establish, using an analytic semigroup approach, short time existence, uniqueness and optimal H\"older regularity in time and space of classical solutions of the nonlocal equation, depending on the regularity of the initial graph. The method also yields $C^\infty-$smoothness estimates of the evolving graphs for positive times.