Paper · Sep 7, 2025

Curvature Flow of Networks with Triple Junction Drag

Yuchuan Yang, Selim Esedoglu

math.APWider flowsarXiv:2509.06125v2
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Abstract

We consider a PDE system that describes curvature motion of networks with a dynamic boundary condition known as triple junction drag. This model arises in the study of grain boundary evolution in polycrystalline materials. In this system, the surface tension coefficients depend on the crystallographic orientations of the grains, which are allowed to rotate. We prove existence and uniqueness of solutions to this system in the parabolic Hölder class . Moreover, we extend our existence result to accommodate a wider class of initial networks by relaxing the compatibility conditions on the angles and curvatures at the triple junction. As an important by-product of our result, we demonstrate the possibility of a new type of topological change during the evolution of the network. We also revisit the question of stability of stationary networks and how it is affected by the choice of surface tensions.

Other curvature flowsAnisotropic & nonlocal flowsStability & uniquenessBoundaries & non-compactPhase 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 Weak-strong uniqueness for the mean curvature flow of double bubbles (Sebastian Hensel and Tim Laux, 2021)8Hensel 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 and Keisuke Takasao, 2021)2Kagaya 2021 Type-0 singularities in the network flow – Evolution of trees (Carlo Mantegazza, Matteo Novaga and Alessandra Pluda, 2022)6Mantegazza 2022 Some aspects of anisotropic curvature flow of planar partitions (Giovanni Bellettini and Shokhrukh Yu. Kholmatov, 2023)7Bellettini 2023 Stability analysis for the anisotropic curve shortening flow of planar networks (Michael Gößwein, Matteo Novaga and Paola Pozzi, 2023)3Gößwein 2023 Crystalline hexagonal curvature flow of networks: short-time, long-time and self-similar evolutions (Giovanni Bellettini, Shokhrukh Kholmatov and Firdavsjon Almuratov, 2024)5Bellettini 2024 Self-Expanding Solutions to the Mean Curvature Flow for Multiphase Surfaces with Regular Junctions (Wei-Hung Liao, 2025)4Liao 2025 A Phase-Field Method for Curvature Flow of Networks with Triple Junction Drag (Yuchuan Yang and Selim Esedoglu, 2026)1Yang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A Phase-Field Method for Curvature Flow of Networks with Triple Junction Drag

    Yuchuan Yang and Selim EsedogluWider flows PDF

    Topics Phase fields & multiphase flows, Anisotropic & nonlocal flows, Other curvature flows · Same authors Yuchuan Yang, Selim Esedoglu · Shared terms “junction drag”, “triple junction”, “model arises”

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

    Takashi Kagaya, Masashi Mizuno and Keisuke TakasaoWider flows PDF

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

  3. Stability analysis for the anisotropic curve shortening flow of planar networks

    Michael Gößwein, Matteo Novaga and Paola PozziWider flows PDF

    Topics Phase fields & multiphase flows, Anisotropic & nonlocal flows, Stability & uniqueness · Shared terms “triple junction”

  4. 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”

  5. 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, Anisotropic & nonlocal flows, Other curvature flows

  6. Type-0 singularities in the network flow – Evolution of trees

    Carlo Mantegazza, Matteo Novaga and Alessandra PludaWider flows PDF

    Topics Phase fields & multiphase flows · Shared terms “junctions”

  7. Some aspects of anisotropic curvature flow of planar partitions

    Giovanni Bellettini and Shokhrukh Yu. KholmatovWider flows PDF

    Topics Anisotropic & nonlocal flows, Other curvature flows · Shared terms “polycrystalline”

  8. Weak-strong uniqueness for the mean curvature flow of double bubbles

    Sebastian Hensel and Tim LauxWider flows PDF

    Topics Phase fields & multiphase flows, Stability & uniqueness · 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) 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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)