Paper · Jun 10, 2022
Consistency of the flat flow solution to the volume preserving mean curvature flow
As on arXiv
Abstract
We consider the flat flow solution, obtained via discrete minimizing movement scheme, to the volume preserving mean curvature flow starting from C^1,1-regular set. We prove the consistency principle which states that (any) such flat flow agrees with the classical solution as long as the latter exists. In particular, the flat flow is unique and smooth up to the first singular time. We obtain the result by proving the full regularity for the discrete time approximation of the flat flow such that the regularity estimates are stable with respect to the time discretization. Our method can also be applied in the case of the mean curvature flow and thus it provides an alternative proof, not relying on comparison principle, for the consistency between the flat flow solution and the classical solution for C^1,1-regular initial sets.
Its neighbourhood
Connected papers
The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.
-
Flat flow solution to the mean curvature flow with volume constraint
Topics Volume-preserving & constrained flows, Weak & singular flows · Same author Vesa Julin · Shared terms “flat flow”, “discrete minimizing”, “movement scheme”
-
Eventual regularity of the volume-preserving mean curvature flow in three and two dimensions
Topics Volume-preserving & constrained flows, Weak & singular flows · Same author Vesa Julin · Shared terms “flat flow”, “discrete minimizing”, “movement scheme”
-
The Asymptotics of the Area-Preserving Mean Curvature and the Mullins-Sekerka Flow in Two Dimensions
Topics Volume-preserving & constrained flows · Same author Vesa Julin · Shared terms “flat flow”
-
De Giorgi varifold solutions to Mean Curvature Flow: a minimizing movements approach
Topics Volume-preserving & constrained flows, Weak & singular flows · Shared terms “minimizing movements”, “comparison principle”, “movement scheme”
-
Weak-strong uniqueness for volume-preserving mean curvature flow
Topics Volume-preserving & constrained flows, Weak & singular flows
-
Existence of flat flows for volume-preserving mean curvature flow with contact angle
Topics Volume-preserving & constrained flows, Weak & singular flows · Shared terms “flat flow”
-
A variational approach to the volume-preserving anisotropic mean curvature flow in 2D
Topics Volume-preserving & constrained flows, Weak & singular flows · Shared terms “classical solution”, “movement scheme”, “minimizing movements”
-
Discrete and Continuum Area-Preserving Mean-Curvature Flow of Rectangles
Topics Volume-preserving & constrained flows, Weak & singular flows · Shared terms “flat flow”, “movement scheme”, “minimizing movements”
Read around it
Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 12C.9After Perelman
Singular flows, the Smale conjecture, and Bamler’s theory.
- 6A.10Curve Shortening and the First Geometric Flows
Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.