Paper · Apr 24, 2018

Convergence of Landau-Lifshitz equation to multi-phase Brakke's mean curvature flow

Keisuke Takasao

math.APWider flowsarXiv:1804.08805
View PDF View HTML arXiv page

As on arXiv

Abstract

We study the convergence of the system of the Allen-Cahn equations to the weak solution for the multi-phase mean curvature flow in the sense of Brakke. The Landau-Lifshitz equation in this paper can be regarded as a system of Allen-Cahn equations with the Lagrange multiplier, which is a phase field model of the multi-phase mean curvature flow. Under an assumption that the limit of the energies of the solutions for the equations matches with the total variation for the singular limit of the solutions, we show that the family of the varifolds derived from the energies is a Brakke flow.

View PDF View HTML arXiv page
Submitted
Apr 24, 2018
Subjects
math.AP

Comments: 13 pages

Mean curvature flowWeak & singular flowsPhase fields & multiphase 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 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 Convergence of the Allen-Cahn Equation to multi-phase mean-curvature flow (Tim Laux and Thilo Simon, 2016)2Laux 2016 A varifold formulation of mean curvature flow with Dirichlet or dynamic boundary conditions (Yoshikazu Giga, Fumihiko Onoue and Keisuke Takasao, 2018)4Giga 2018 Existence of weak solution for mean curvature flow with transport term and forcing term (Keisuke Takasao, 2019)7Takasao 2019 On obstacle problem for Brakke's mean curvature flow (Keisuke Takasao, 2021)1Takasao 2021 A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness (Sebastian Hensel and Tim Laux, 2021)6Hensel 2021 Existence of weak solution to volume preserving mean curvature flow in higher dimensions (Keisuke Takasao, 2022)8Takasao 2022 Brakke inequality and the existence of Brakke-flow for volume preserving mean curvature flow (Andrea Chiesa and Keisuke Takasao, 2025)5Chiesa 2025 Convergence of a vector-valued Allen-Cahn system to Brakke's multiphase mean curvature flow (Tim Laux and Keisuke Takasao, 2026)3Laux 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On obstacle problem for Brakke's mean curvature flow

    Keisuke TakasaoWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows · Same author Keisuke Takasao · Shared terms “Brakke mean”, “Allen Cahn”, “Cahn equation”

  2. Convergence of the Allen-Cahn Equation to multi-phase mean-curvature flow

    Tim Laux and Thilo SimonWider flows

    Topics Phase fields & multiphase flows · Shared terms “multi phase”, “Allen Cahn”, “Cahn equation”

  3. Convergence of a vector-valued Allen-Cahn system to Brakke's multiphase mean curvature flow

    Tim Laux and Keisuke TakasaoWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows · Same author Keisuke Takasao · Shared terms “Allen Cahn”, “Brakke”, “Cahn equation”

  4. A varifold formulation of mean curvature flow with Dirichlet or dynamic boundary conditions

    Yoshikazu Giga, Fumihiko Onoue and Keisuke TakasaoWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows · Same author Keisuke Takasao · Shared terms “varifolds”, “Allen Cahn”, “Cahn equation”

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

    Andrea Chiesa and Keisuke TakasaoWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows · Same author Keisuke Takasao · Shared terms “Brakke”, “varifolds”

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

    Sebastian Hensel and Tim LauxWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows · Shared terms “multi phase”, “Allen Cahn”, “Cahn equation”

  7. Existence of weak solution for mean curvature flow with transport term and forcing term

    Keisuke TakasaoWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows · Same author Keisuke Takasao · Shared terms “Allen Cahn”, “Cahn equation”

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

    Keisuke TakasaoWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows · Same author Keisuke Takasao · Shared terms “Cahn equation”, “Allen”, “varifolds”

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) 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) Phase fields & multiphase flows Interfaces moving by curvature can be approximated by smooth phase fields: the Allen–Cahn equation's transition layers converge to mean curvature… Start with: Two-Dimensional Motion of Idealized Grain Boundaries (Mullins, 1956)

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.