Weak solutions of Mullins-Sekerka flow as a Hilbert space gradient flow

TL;DR

This paper introduces a new weak solution framework for the Mullins-Sekerka flow based on a gradient flow perspective, incorporating energy dissipation and contact angle conditions, and proves existence via a minimizing movements scheme.

Contribution

It develops a novel weak solution theory for Mullins-Sekerka flow that includes energy dissipation and contact angle conditions, unifying previous approaches.

Findings

Established existence of weak solutions through a minimizing movements scheme.

Ensured consistency with classical smooth solutions.

Provided a framework potentially compatible with relative entropy methods.

Abstract

We propose a novel weak solution theory for the Mullins-Sekerka equation primarily motivated from a gradient flow perspective. Previous existence results on weak solutions due to Luckhaus and Sturzenhecker (Calc. Var. PDE 3, 1995) or R\"oger (SIAM J. Math. Anal. 37, 2005) left open the inclusion of both a sharp energy dissipation principle and a weak formulation of the contact angle at the intersection of the interface and the domain boundary. To incorporate these, we introduce a functional framework encoding a weak solution concept for Mullins-Sekerka flow essentially relying only on (i) a single sharp energy dissipation inequality in the spirit of De~Giorgi, and (ii) a weak formulation for an arbitrary fixed contact angle through a distributional representation of the first variation of the underlying capillary energy. Both ingredients are intrinsic to the interface of the evolving phase indicator and an explicit distributional PDE formulation with potentials can be derived from them. Existence of weak solutions is established via subsequential limit points of the naturally associated minimizing movements scheme. Smooth solutions are consistent with the classical Mullins-Sekerka flow, and even further, we expect our solution concept to be amenable, at least in principle, to the recently developed relative entropy approach for curvature driven interface evolution.