Mullins-Sekerka as the Wasserstein flow of the perimeter

TL;DR

This paper demonstrates that the Mullins-Sekerka equation can be viewed as a Wasserstein gradient flow of the perimeter, establishing convergence of a related implicit time discretization method.

Contribution

It introduces a novel interpretation of the Mullins-Sekerka equation as a Wasserstein flow and proves convergence of a discretization scheme using optimal transport and geometric measure theory.

Findings

Convergence of the implicit time discretization for the Mullins-Sekerka equation.

The limit satisfies the equation in a distributional sense.

The approach combines optimal transport, gradient flows, and geometric measure theory.

Abstract

We prove the convergence of an implicit time discretization for the one-phase Mullins-Sekerka equation, possibly with additional non-local repulsion, proposed in [F. Otto, Arch. Rational Mech. Anal. 141 (1998) 63--103]. Our simple argument shows that the limit satisfies the equation in a distributional sense as well as an optimal energy-dissipation relation. The proof combines arguments from optimal transport, gradient flows & minimizing movements, and basic geometric measure theory.