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

@article{Laux2022WeakstrongUF, title={Weak-strong uniqueness for volume-preserving mean curvature flow}, author={Tim Laux}, journal={Revista Matem{\'a}tica Iberoamericana}, year={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, which is a natural extension of the concept in the case without volume preservation recently introduced by Fischer et al. [arXiv:2003.05478]. The ﬁrst main result shows that any strong solution with certain regularity is calibrated. The second main result consists of a stability estimate in terms…

## 4 Citations

### 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…

### A uniqueness and stability principle for surface diffusion

- Mathematics
- 2022

. We derive a uniqueness and stability principle for surface diﬀusion before the onset of singularities. The perturbations, however, are allowed to undergo topological changes. The main ingredient is…

### Flat flow solution to the mean curvature flow with volume constraint

- Geology
- 2023

. In this paper I will revisit the construction of a global weak solution to the volume preserving mean curvature ﬂow via discrete minimizing movement scheme by Mugnai-Seis-Spadaro [19]. This method…

### Diffuse-interface approximation and weak-strong uniqueness of anisotropic mean curvature flow

- Mathematics
- 2022

The purpose of this paper is to derive anisotropic mean curvature flow as the limit of the anisotropic Allen–Cahn equation. We rely on distributional solution concepts for both the diffuse and sharp…

## References

SHOWING 1-10 OF 37 REFERENCES

### Global solutions to the volume-preserving mean-curvature flow

- Mathematics
- 2015

In this paper, we construct global distributional solutions to the volume-preserving mean-curvature flow using a variant of the time-discrete gradient flow approach proposed independently by Almgren…

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

- Mathematics
- 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…

### Existence of weak solution to volume preserving mean curvature flow in higher dimensions

- Mathematics
- 2022

Abstract. In this paper, we construct a family of integral varifolds, which is a global weak solution to the volume preserving mean curvature flow in the sense of L-flow. This flow is also a…

### Existence of weak solution for volume preserving mean curvature flow via phase field method

- Mathematics
- 2015

We study the phase field method for the volume preserving mean curvature flow. Given an initial $C^1$ hypersurface we proved the existence of the weak solution for the volume preserving mean…

### Convergence of thresholding schemes incorporating bulk effects

- Computer Science
- 2016

The convergence of three computational algorithms for interface motion in a multi-phase system, which incorporate bulk effects, is established, including a thresholding scheme for simulating grain growth in a polycrystal surrounded by air.

### Convergence of the Allen‐Cahn Equation to Multiphase Mean Curvature Flow

- Mathematics
- 2016

We present a convergence result for solutions of the vector‐valued Allen‐Cahn equation. In the spirit of the work of Luckhaus and Sturzenhecker we establish convergence towards a distributional…

### 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…

### Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow

- MathematicsArXiv
- 2022

. Phase-ﬁeld models such as the Allen-Cahn equation may give rise to the formation and evolution of geometric shapes, a phenomenon that may be analyzed rigorously in suitable scaling regimes. In its…

### Long Time Behaviour of the Discrete Volume Preserving Mean Curvature Flow in the Flat Torus

- Mathematics
- 2022

We show that the discrete approximate volume preserving mean curvature ﬂow in the ﬂat torus T N starting near a strictly stable critical set E of the perimeter converges in the long time to a…

### The asymptotics of the area-preserving mean curvature and the Mullins–Sekerka flow in two dimensions

- MathematicsMathematische Annalen
- 2022

We provide the first general result for the asymptotics of the area preserving mean curvature flow in two dimensions showing that flat flow solutions, starting from any bounded set of finite…