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

2024/04/03 by Julian Fischer, Sebastian Hensel, Fischer, Julian +5 · 2 citations
Mathematics · Engineering · #Advanced Mathematical Physics Problems #Stability and Controllability of Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2404.02682

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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• Convergence to equilibrium of weak solutions to the Cahn--Hilliard equation with non-degenerate mobility and singular potential
• Varifold solutions to Volume-Preserving Mean Curvature Flow: existence and weak-strong uniqueness

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