Paper · Jun 7, 2018

On the self-similar solutions of the crystalline mean curvature flow in three dimensions

Norbert Požár

math.APWider flowsarXiv:1806.02482
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Abstract

We present two types of self-similar shrinking solutions of positive genus for the crystalline mean curvature flow in three dimensions analogous to the solutions known for the standard mean curvature flow. We use them to test a numerical implementation of a level set algorithm for the crystalline mean curvature flow in three dimensions based on the minimizing movements scheme of A. Chambolle, Interfaces Free Bound. 6 (2004). We implement a finite element method discretization that seems to improve the handling of edges in three dimensions compared to the standard finite difference method and illustrate its behavior on a few examples.

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Submitted
Jun 7, 2018
Subjects
math.AP, math.NA

Comments: 28 pages, 18 figures

Mean curvature flowAnisotropic & nonlocal flowsWeak & singular flowsNumerics & visualization

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 2016, 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 A level set crystalline mean curvature flow of surfaces (Yoshikazu Giga and Norbert Požár, 2016)1Giga 2016 Existence and uniqueness for anisotropic and crystalline mean curvature flows (Antonin Chambolle, Massimiliano Morini, Matteo Novaga et al., 2017)5Chambolle 2017 Approximation of general facets by regular facets with respect to anisotropic total variation energies and its application to the crystalline mean curvature flow (Yoshikazu Giga and Norbert Požár, 2017)2Giga 2017 Anisotropic and crystalline mean curvature flow of mean-convex sets (Antonin Chambolle and Matteo Novaga, 2020)7Chambolle 2020 Viscosity solutions for the crystalline mean curvature flow with a nonuniform driving force term (Yoshikazu Giga and Norbert Pozar, 2020)3Giga 2020 On volume-preserving crystalline mean curvature flow (Inwon Kim, Dohyun Kwon and Norbert Požár, 2020)4Kim 2020 Graphical translators for anisotropic and crystalline mean curvature flow (Annalisa Cesaroni, Heiko Kroener and Matteo Novaga, 2021)8Cesaroni 2021 Motion by crystalline-like mean curvature: a survey (Yoshikazu Giga and Norbert Požár, 2021)6Giga 2021
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A level set crystalline mean curvature flow of surfaces

    Yoshikazu Giga and Norbert PožárWider flows

    Topics Anisotropic & nonlocal flows, Weak & singular flows · Same author Norbert Požár · Shared terms “crystalline mean”, “level set”

  2. Approximation of general facets by regular facets with respect to anisotropic total variation energies and its application to the crystalline mean curvature flow

    Yoshikazu Giga and Norbert PožárWider flows

    Topics Anisotropic & nonlocal flows, Weak & singular flows · Same author Norbert Požár · Shared terms “crystalline mean”, “level set”

  3. Viscosity solutions for the crystalline mean curvature flow with a nonuniform driving force term

    Yoshikazu Giga and Norbert PozarWider flows

    Topics Anisotropic & nonlocal flows, Weak & singular flows · Same author Norbert Požár · Shared terms “crystalline mean”, “level set”

  4. On volume-preserving crystalline mean curvature flow

    Inwon Kim, Dohyun Kwon and Norbert PožárWider flows

    Topics Anisotropic & nonlocal flows, Weak & singular flows · Same author Norbert Požár · Shared terms “crystalline mean”

  5. Existence and uniqueness for anisotropic and crystalline mean curvature flows

    Antonin Chambolle, Massimiliano Morini, Matteo Novaga et al.Wider flows

    Topics Anisotropic & nonlocal flows, Weak & singular flows · Shared terms “crystalline mean”, “movements scheme”, “minimizing movements”

  6. Motion by crystalline-like mean curvature: a survey

    Yoshikazu Giga and Norbert PožárWider flows

    Topics Anisotropic & nonlocal flows · Same author Norbert Požár · Shared terms “crystalline”

  7. Anisotropic and crystalline mean curvature flow of mean-convex sets

    Antonin Chambolle and Matteo NovagaWider flows

    Topics Anisotropic & nonlocal flows · Shared terms “crystalline mean”

  8. Graphical translators for anisotropic and crystalline mean curvature flow

    Annalisa Cesaroni, Heiko Kroener and Matteo NovagaWider flows

    Topics Anisotropic & nonlocal flows · Shared terms “crystalline mean”

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 in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Anisotropic & nonlocal flows Wider flows When surface energy depends on direction, as in crystals, mean curvature flow becomes anisotropic, and the Wulff shape replaces the round sphere. Start with: Crystalline variational problems (Taylor, 1978) 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) Numerics & visualization Computing and visualising Ricci flow: simulations of neckpinches and critical behaviour, pictures of rotationally symmetric flows, and… Start with: Deforming metrics in the direction of their Ricci tensors (DeTurck, 1983)

In the guidebook

  1. 6D.11
    Computing Geometric FlowsBook 6D · Diffusions, Finance and Computation

    Numerical curve shortening and mean curvature flow, level sets and phase fields, discrete Ricci flow on surfaces and in conformal mapping, rotationally symmetric neckpinches, and what computation has taught about singularities.

  2. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

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

  3. 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.