Paper · Dec 21, 2021

Long time behavior for a curvature flow of networks related to grain bundary motion with the effect of lattice misorientations

Takashi Kagaya, Masashi Mizuno, Keisuke Takasao

math.APWider flowsarXiv:2112.11069v3
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Abstract

The mathematical model of grain boundary motion, including lattice misorientations' effect, is considered. When time-dependent lattice misorientations are state variables of the surface tension of the grain boundary, to ensure the energy dissipation law, one can obtain a curvature flow of networks with time-dependent mobilities. This paper studies the solvability and long-time asymptotic behavior of the curvature flow subjected to the Herring condition which ensures that the constituent grain boundary surface tensions are balanced at the triple junction.

Comments: 58 pages, 1 figure

Other curvature flowsLong-time existence & convergencePhase fields & multiphase flows

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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 occurrence of surface tension gradient discontinuities and zero mobility for Allen-Cahn and curvature flows in periodic media (William M Feldman and Peter S Morfe, 2021)3Feldman 2021 Existence of weak solution to volume preserving mean curvature flow in higher dimensions (Keisuke Takasao, 2022)5Takasao 2022 Crystalline hexagonal curvature flow of networks: short-time, long-time and self-similar evolutions (Giovanni Bellettini, Shokhrukh Kholmatov and Firdavsjon Almuratov, 2024)6Bellettini 2024 Brakke inequality and the existence of Brakke-flow for volume preserving mean curvature flow (Andrea Chiesa and Keisuke Takasao, 2025)4Chiesa 2025 Curvature Flow of Networks with Triple Junction Drag (Yuchuan Yang and Selim Esedoglu, 2025)1Yang 2025 Self-Expanding Solutions to the Mean Curvature Flow for Multiphase Surfaces with Regular Junctions (Wei-Hung Liao, 2025)8Liao 2025 Convergence of a vector-valued Allen-Cahn system to Brakke's multiphase mean curvature flow (Tim Laux and Keisuke Takasao, 2026)2Laux 2026 Global weak varifold solutions to Navier-Stokes/Mean Curvature Flow system (Andrea Poiatti, 2026)7Poiatti 2026
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  1. Curvature Flow of Networks with Triple Junction Drag

    Yuchuan Yang and Selim EsedogluWider flows PDF

    Topics Phase fields & multiphase flows, Other curvature flows · Shared terms “grain boundary”, “surface tension”, “triple junction”

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

    Tim Laux and Keisuke TakasaoWider flows PDF

    Topics Phase fields & multiphase flows · Same author Keisuke Takasao

  3. The occurrence of surface tension gradient discontinuities and zero mobility for Allen-Cahn and curvature flows in periodic media

    William M Feldman and Peter S MorfeWider flows PDF

    Topics Phase fields & multiphase flows, Other curvature flows · Shared terms “surface tension”, “mobility”

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

    Andrea Chiesa and Keisuke TakasaoWider flows PDF

    Topics Phase fields & multiphase flows · Same author Keisuke Takasao

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

    Keisuke TakasaoWider flows PDF

    Topics Phase fields & multiphase flows · Same author Keisuke Takasao

  6. Crystalline hexagonal curvature flow of networks: short-time, long-time and self-similar evolutions

    Giovanni Bellettini, Shokhrukh Kholmatov and Firdavsjon AlmuratovWider flows PDF

    Topics Phase fields & multiphase flows, Other curvature flows · Shared terms “long time”

  7. Global weak varifold solutions to Navier-Stokes/Mean Curvature Flow system

    Andrea PoiattiWider flows PDF

    Topics Phase fields & multiphase flows, Long-time existence & convergence · Shared terms “surface tension”, “energy dissipation”, “mobility”

  8. Self-Expanding Solutions to the Mean Curvature Flow for Multiphase Surfaces with Regular Junctions

    Wei-Hung LiaoWider flows PDF

    Topics Phase fields & multiphase flows · Shared terms “triple junction”

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Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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)