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A weak-strong uniqueness principle for the Mullins-Sekerka equation

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arxiv 2404.02682 v2 pith:ULOKT62K submitted 2024-04-03 math.AP

Julian Fischer , Sebastian Hensel , Tim Laux , Theresa M. Simon This is my paper · ORCID
classification math.AP
keywords equationmullins-sekerkaevolutionnotionperturbativeprinciplesolutionstability
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abstract

We establish a weak-strong uniqueness principle for the two-phase Mullins-Sekerka equation in ambient dimension $d = 2$ and $3$: As long as a classical solution to the evolution problem exists, any weak De Giorgi type varifold solution (see for this notion the recent work of Stinson and the second author, Arch. Ration. Mech. Anal. 248, 8, 2024) must coincide with it. In particular, in the absence of geometric singularities such weak solutions do not introduce a mechanism for (unphysical) non-uniqueness. We also derive a stability estimate with respect to changes in the data. Our method is based on the notion of relative entropies for interface evolution problems, a reduction argument to a perturbative graph setting, and a stability analysis in this perturbative regime relying crucially on the gradient flow structure of the Mullins-Sekerka equation.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. De Giorgi varifold solutions to Mean Curvature Flow: a minimizing movements approach

    math.AP 2026-07 accept novelty 7.5 of 10

    A new Mullins–Sekerka-regularized proxy for the degenerate L² distance yields the first unconditional minimizing-movements convergence to De Giorgi varifold solutions of mean curvature flow.

  2. Existence of weak solutions to volume-preserving mean curvature flow with obstacles

    math.AP 2025-01 conditional novelty 6.0 of 10

    Global weak solutions to volume-preserving mean curvature flow with obstacles exist in all dimensions by Allen-Cahn phase-field approximation.

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