Global Existence and Stability for the Modified Mullins-Sekerka and Surface Diffusion Flow

TL;DR

This survey reviews the asymptotic behavior and stability of the modified Mullins-Sekerka and surface diffusion flows, emphasizing the role of the nonlocal Area functional and strict stability in ensuring global existence and convergence.

Contribution

It establishes the equivalence of strict stability and minimality under perturbations and demonstrates global existence and convergence of flows near stable critical sets in low dimensions.

Findings

Flows exist globally for initial sets close to stable critical sets.

Flows asymptotically converge to a translate of the critical set.

Strict stability is necessary and sufficient for minimality under perturbations.

Abstract

In this survey we present the state of the art about the asymptotic behavior and stability of the modified Mullins--Sekerka flow and the surface diffusion flow of smooth sets, mainly due to E.~Acerbi, N.~Fusco, V.Julin and M.Morini. First we discuss in detail the properties of the nonlocal Area functional under a volume constraint, of which the two flows are the gradient flow with respect to suitable norms, in particular, we define the strict stability property for a critical set of such functional and we show that it is a necessary and sufficient condition for minimality under $W^{2,p}$-perturbations, holding in any dimension. Then, we show that, in dimensions two and three, for initial sets sufficiently "close" to a smooth strictly stable critical set $E$, both flows exist for all positive times and asymptotically "converge" to a translate of $E$.