# A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness

@inproceedings{Hensel2021ANV, title={A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness}, author={Sebastian Hensel and Tim Laux}, year={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, just as in Brakke’s formulation, but are coupled to the phase volumes by a simple transport equation. First, we show that, in the exact same setup as in Ilmanen’s proof [J. Differential Geom. 38, 417–461, (1993)], any limit point of solutions to the Allen–Cahn equation is a varifold solution in our…

## 9 Citations

### On the existence of canonical multi-phase Brakke flows

- MathematicsAdvances in Calculus of Variations
- 2022

Abstract 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…

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

- Mathematics
- 2022

. We propose a novel weak solution theory for the Mullins–Sekerka equation primarily motivated from a gradient ﬂow perspective. Previous ex- istence results on weak solutions due to Luckhaus and…

### Weak-strong uniqueness for volume-preserving mean curvature flow

- MathematicsRevista Matemática Iberoamericana
- 2022

. In this note, we derive a stability and weak-strong uniqueness principle for volume-preserving mean curvature ﬂow. The proof is based on a new notion of volume-preserving gradient ﬂow calibrations,…

### Weak-strong Uniqueness for the Navier–Stokes Equation for Two Fluids with Ninety Degree Contact Angle and Same Viscosities

- MathematicsJournal of Mathematical Fluid Mechanics
- 2022

We consider the flow of two viscous and incompressible fluids within a bounded domain modeled by means of a two-phase Navier–Stokes system. The two fluids are assumed to be immiscible, meaning that…

### The sharp interface limit of a Navier--Stokes/Allen--Cahn system with constant mobility: Convergence rates by a relative energy approach

- Mathematics
- 2022

. We investigate the sharp interface limit of a diﬀuse interface sys- tem that couples the Allen–Cahn equation with the instationary Navier–Stokes system in a bounded domain in R d with d ∈ { 2 , 3 }…

### Consistency of the flat flow solution to the volume preserving mean curvature flow

- Mathematics
- 2022

We consider the flat flow solution, obtained via discrete minimizing movement scheme, to the volume preserving mean curvature flow starting from C-regular set. We prove the consistency principle…

### Convergence rates for the Allen–Cahn equation with boundary contact energy: the non-perturbative regime

- Mathematics, Computer ScienceCalculus of Variations and Partial Differential Equations
- 2022

The present work removes the perturbative assumption on the contact angle being close to 90 and establishes under usual double-well type assumptions on the potential and for a certain class of boundary energy densities the sub-optimal convergence rate of order.

### Sharp Interface Limit of the Cahn-Hilliard Reaction Model for Lithium-ion Batteries

- Mathematics
- 2022

. We propose a weak solution theory for the sharp interface limit of the Cahn–Hilliard reaction model, a variational PDE for lithium-ion batteries. An essential feature of this model is the use of…

### De Giorgi’s inequality for the thresholding scheme with arbitrary mobilities and surface tensions

- MathematicsCalculus of Variations and Partial Differential Equations
- 2022

We provide a new convergence proof of the celebrated Merriman–Bence–Osher scheme for multiphase mean curvature flow. Our proof applies to the new variant incorporating a general class of surface…

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