We consider the volume-preserving geometric evolution of the boundary of a set under fractional mean curvature. We show that smooth convex solutions maintain their fractional curvatures bounded for all times, and the long-time asymptotics approach round spheres. The proofs are based on a priori estimates on the inner and outer radii of the solutions.
Eleonora Cinti. Carlo Sinestrari. Enrico Valdinoci. "Convex sets evolving by volume-preserving fractional mean curvature flows." Anal. PDE 13 (7) 2149 - 2171, 2020. https://doi.org/10.2140/apde.2020.13.2149