Nonlinear stability results for the modified Mullins-Sekerka and the surface diffusion flow

TL;DR

This paper proves exponential stability of certain three-dimensional periodic configurations under surface diffusion, Mullins-Sekerka, and Hele-Shaw flows, based on their stability for the area and Ohta-Kawasaki energies.

Contribution

It establishes the exponential stability of strictly stable configurations for these flows, extending stability results to the modified Mullins-Sekerka flow.

Findings

Strictly stable configurations are exponentially stable under surface diffusion and Mullins-Sekerka flows.

Stability results extend to configurations stable for the Ohta-Kawasaki energy.

Results apply to three-dimensional periodic configurations.

Abstract

It is shown that any three-dimensional periodic configuration that is strictly stable for the area functional is exponentially stable for the surface diffusion flow and for the Mullins-Sekerka or Hele-Shaw flow. The same result holds for three-dimensional periodic configurations that are strictly stable with respect to the sharp-interface Ohta-Kawaski energy. In this case, they are exponentially stable for the so-called modified Mullins-Sekerka flow.