The local structure of the energy landscape in multiphase mean curvature flow: Weak-strong uniqueness and stability of evolutions

2020/03/11 by Julian Fischer, Fischer, Julian, Sebastian Hensel +5 · 2 citations
Mathematics · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Black Holes and Theoretical Physics

paper · pdf · doi:10.48550/arxiv.2003.05478

Abstract

We prove that in the absence of topological changes, the notion of BV solutions to planar multiphase mean curvature flow does not allow for a mechanism for (unphysical) non-uniqueness. Our approach is based on the local structure of the energy landscape near a classical evolution by mean curvature. Mean curvature flow being the gradient flow of the surface energy functional, we develop a gradient-flow analogue of the notion of calibrations. Just like the existence of a calibration guarantees that one has reached a global minimum in the energy landscape, the existence of a "gradient flow calibration" ensures that the route of steepest descent in the energy landscape is unique and stable.

Citations

• Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations
• The second law of thermodynamics and stability
• Metric spaces and positive definite functions
• Evolution of networks with multiple junctions
• A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness

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• Equivalence of weak solution concepts for mean curvature flow
• Varifold solutions to Volume-Preserving Mean Curvature Flow: existence and weak-strong uniqueness

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