Corpus ID: 212675755

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

@article{Fischer2020TheLS,
title={The local structure of the energy landscape in multiphase mean curvature flow: Weak-strong uniqueness and stability of evolutions},
author={Julian Fischer and Sebastian Hensel and Tim Laux and Thilo M. Simon},
journal={arXiv: Analysis of PDEs},
year={2020}
}
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… Expand

#### Figures and Tables from this paper

On the existence of canonical multi-phase Brakke flows
• Mathematics
• 2021
This paper establishes the global-in-time existence of a multi-phase mean curvature flow, evolving from an arbitrary closed rectifiable initial datum, which is a Brakke flow and a BV solution at theExpand
A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness
• Mathematics
• 2021
We propose a new weak solution concept for (two-phase) mean curvature flow which enjoys both (unconditional) existence and (weak-strong) uniqueness properties. These solutions are evolving varifolds,Expand
Weak-strong uniqueness for the mean curvature flow of double bubbles
• Mathematics
• 2021
We derive a weak-strong uniqueness principle for BV solutions to multiphase mean curvature flow of triple line clusters in three dimensions. Our proof is based on the explicit construction of aExpand
De Giorgi's inequality for the thresholding scheme with arbitrary mobilities and surface tensions
• Mathematics
• 2021
Abstract. We provide a new convergence proof of the celebrated MerrimanBence-Osher scheme for multiphase mean curvature flow. Our proof applies to the new variant incorporating a general class ofExpand
Distributional solutions to mean curvature flow
These lecture notes aim to present some of the ideas behind the recent (conditional) existence and (weak-strong) uniqueness theory for mean curvature flow. Focusing on the simplest case of theExpand
Short-time existence for the network flow
• Mathematics
• 2021
This paper contains a new proof of the short-time existence for the flow by curvature of a network of curves in the plane. Appearing initially in metallurgy and as a model for the evolution of grainExpand
Convergence Rates of the Allen-Cahn Equation to Mean Curvature Flow: A Short Proof Based on Relative Entropies
• Mathematics, Computer Science
• SIAM J. Math. Anal.
• 2020
We give a short and self-contained proof for rates of convergence of the Allen-Cahn equation towards mean curvature flow, assuming that a classical (smooth) solution to the latter exists and startingExpand
Existence and regularity theorems of one-dimensional Brakke flows
• Mathematics
• 2020
Given a closed countably $1$-rectifiable set in $\mathbb R^2$ with locally finite $1$-dimensional Hausdorff measure, we prove that there exists a Brakke flow starting from the given set with theExpand

#### References

SHOWING 1-10 OF 50 REFERENCES
Evolution of networks with multiple junctions
• Mathematics
• 2016
We consider the motion by curvature of a network of curves in the plane and we discuss existence, uniqueness, singularity formation and asymptotic behavior of the flow.
The motion of a surface by its mean curvature
The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press to preserve the original texts of these important books while presenting them in durable paperback editions. Expand
The grain boundary mobility tensor
• Physics, Medicine
• Proceedings of the National Academy of Sciences
• 2020
It is argued that the mobility is, in general, a tensor (classically, it is a scalar) and determine all of its components and demonstrated that stress generation during GB migration necessarily slows grain growth and reduces GB mobility in polycrystals. Expand
Weak-strong uniqueness for multiphase mean curvature flow of double bubbles
• in preparation,
• 2020
Calibrations and null-Lagrangians for nonlocal perimeters and an application to the viscosity theory
For nonnegative even kernels K , we consider the K -nonlocal perimeter functional acting on sets. Assuming the existence of a foliation of space made of solutions of the associated K -nonlocal meanExpand
Halfspaces minimise nonlocal perimeter: a proof via calibrations
We consider a nonlocal functional $J_K$ that may be regarded as a nonlocal version of the total variation. More precisely, for any measurable function $u\colon \mathbb{R}^d \to \mathbb{R}$, we defineExpand
On short time existence for the planar network flow
• Mathematics
• Journal of Differential Geometry
• 2019
We prove the existence of the flow by curvature of regular planar networks starting from an initial network which is non-regular. The proof relies on a monotonicity formula for expanding solutionsExpand
Weak–Strong Uniqueness for the Navier–Stokes Equation for Two Fluids with Surface Tension
• Physics, Mathematics
• 2019
In the present work, we consider the evolution of two fluids separated by a sharp interface in the presence of surface tension—like, for example, the evolution of oil bubbles in water. Our mainExpand
Grain-boundary kinetics: A unified approach
• Materials Science, Physics
• Progress in Materials Science
• 2018
Grain boundaries (GBs) are central defects for describing polycrystalline materials, and playing major role in a wide-range of physical properties of polycrystals. Control over GB kinetics providesExpand
Brakke’s inequality for the thresholding scheme
• Mathematics
• 2017
We continue our analysis of the thresholding scheme from the variational viewpoint and prove a conditional convergence result towards Brakke’s notion of mean curvature flow. Our proof is based on aExpand