Paper · May 21, 2023

Existence of BV flow via elliptic regularization

Kiichi Tashiro

math.DGWider flowsarXiv:2305.12374v2Hiroshima Math. J. 54 (2) 233 - 259, (2024)
PDF Abstract page

As on arXiv

Abstract

We investigate a mean curvature flow obtained via elliptic regularization, and prove that it is not only a Brakke flow, but additionally a generalized BV flow proposed by Stuvard and Tonegawa. In particular, we show that the change in volume of the evolving phase can be expressed in terms of the generalized mean curvature of the Brakke flow.

Comments: 19 pages, I fixed minor mistakes. To appear in Hiroshima math. J

Mean curvature flowWeak & singular flows

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.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper The Parabolic U(1)-Higgs Equations and Codimension-two Mean Curvature Flows (Davide Parise, Alessandro Pigati and Daniel Stern, 2023)3Parise 2023 Existence of weak mean curvature flow with prescribed contact angle via elliptic regularization (Kiichi Tashiro, 2024)1Tashiro 2024 Brakke inequality and the existence of Brakke-flow for volume preserving mean curvature flow (Andrea Chiesa and Keisuke Takasao, 2025)7Chiesa 2025 On the avoidance principle for codimension 1 space-time Brakke flows (Abdelmouksit Sagueni, 2025)6Sagueni 2025 Approximate mean curvature flows of a general varifold, and their limit spacetime Brakke flow (Blanche Buet, Gian Paolo Leonardi, Simon Masnou et al., 2025)5Buet 2025 Spacelike Brakke flows with boundary in pseudo-Euclidean space (Javier Hervás, 2026)4Hervás 2026 A note on generalized BV mean curvature flow with a critical forcing term (Kiichi Tashiro, 2026)2Tashiro 2026 Generic Mean Curvature Flow With Rough Weights (Fabius Krämer, Tim Laux and Keisuke Takasao, 2026)8Krämer 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Existence of weak mean curvature flow with prescribed contact angle via elliptic regularization

    Kiichi TashiroWider flows PDF

    Topics Weak & singular flows · Same author Kiichi Tashiro · Shared terms “elliptic regularization”, “Brakke”

  2. A note on generalized BV mean curvature flow with a critical forcing term

    Kiichi TashiroWider flows PDF

    Topics Weak & singular flows · Same author Kiichi Tashiro · Shared terms “Brakke flows”, “Tonegawa”, “Stuvard”

  3. The Parabolic -Higgs Equations and Codimension-two Mean Curvature Flows

    Davide Parise, Alessandro Pigati and Daniel SternWider flows PDF

    Topics Weak & singular flows · Shared terms “elliptic regularization”, “Brakke flows”

  4. Spacelike Brakke flows with boundary in pseudo-Euclidean space

    Javier HervásWider flows PDF

    Topics Weak & singular flows · Shared terms “Brakke flows”, “elliptic regularization”, “generalized mean”

  5. Approximate mean curvature flows of a general varifold, and their limit spacetime Brakke flow

    Blanche Buet, Gian Paolo Leonardi, Simon Masnou et al.Wider flows PDF

    Topics Weak & singular flows · Shared terms “Brakke flows”, “generalized mean”, “Tonegawa”

  6. On the avoidance principle for codimension 1 space-time Brakke flows

    Abdelmouksit SagueniWider flows PDF

    Topics Weak & singular flows · Shared terms “Brakke flows”

  7. Brakke inequality and the existence of Brakke-flow for volume preserving mean curvature flow

    Andrea Chiesa and Keisuke TakasaoWider flows PDF

    Topics Weak & singular flows · Shared terms “Brakke flows”

  8. Generic Mean Curvature Flow With Rough Weights

    Fabius Krämer, Tim Laux and Keisuke TakasaoWider flows PDF

    Topics Weak & singular flows · Shared terms “Brakke flows”

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.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002)

In the guidebook

  1. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  2. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.