Paper · Sep 23, 2019

Global stability of traveling waves for an area preserving curvature flow with contact angle condition

Takashi Kagaya

math.APWider flowsarXiv:1909.10150
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Abstract

We consider an evolving plane curve with two endpoints that can move freely on the -axis with generating constant contact angles. We discuss the asymptotic behavior of global-in-time solutions when the evolution of this plane curve is governed by area-preserving curvature flow equation. The main result shows that any moving curve converges to a traveling wave if the moving curve starts from an embedded convex curve and remains bounded in global time.

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Submitted
Sep 23, 2019
Subjects
math.AP, math.DG
Journal
Journal of Differential Equations, Vol. 269, Issue 4, pp. 3489-3514, 2020
DOI
10.1016/j.jde.2020.03.006

Comments: 24pages, 3 figures

Other curvature flowsVolume-preserving & constrained flowsStability & uniquenessFree boundary & capillaryLong-time existence & convergence

Its neighbourhood

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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 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 Nonparametric mean curvature type flows of graphs with contact angle conditions (Hengyu Zhou, 2017)4Zhou 2017 Hypersurfaces with capillary boundary evolving by volume preserving power mean curvature flow (Carlo Sinestrari and Liangjun Weng, 2024)3Sinestrari 2024 Asymptotic circularity of immortal area-preserving curvature flows (Tatsuya Miura, 2024)8Miura 2024 On a non-local area-preserving curve flow (Zezhen Sun and Yuting Wu, 2025)5Sun 2025 The asymptotic of the Mullins-Sekerka and the area-preserving curvature flow in the planar flat torus (Vedansh Arya, Daniele De Gennaro and Anna Kubin, 2025)7Arya 2025 On an area-preserving inverse curvature flow for plane curves (Zezhen Sun and Yuting Wu, 2025)6Sun 2025 Existence of flat flows for volume-preserving mean curvature flow with contact angle (Jiwoong Jang, 2025)2Jang 2025 Area-preserving curvature flow for open curves with general contact angles on skew lines (Fuya Hiroi, 2026)1Hiroi 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Area-preserving curvature flow for open curves with general contact angles on skew lines

    Fuya HiroiWider flows

    Topics Free boundary & capillary, Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “contact angle”, “area preserving”, “angle condition”

  2. Existence of flat flows for volume-preserving mean curvature flow with contact angle

    Jiwoong JangWider flows

    Topics Free boundary & capillary, Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “contact angle”, “angle condition”

  3. Hypersurfaces with capillary boundary evolving by volume preserving power mean curvature flow

    Carlo Sinestrari and Liangjun WengWider flows

    Topics Free boundary & capillary, Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “area preserving”

  4. Nonparametric mean curvature type flows of graphs with contact angle conditions

    Hengyu ZhouWider flows

    Topics Free boundary & capillary, Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “contact angle”, “angle condition”, “asymptotic behavior”

  5. On a non-local area-preserving curve flow

    Zezhen Sun and Yuting WuWider flows

    Topics Volume-preserving & constrained flows, Other curvature flows · Shared terms “area preserving”, “curves converge”

  6. On an area-preserving inverse curvature flow for plane curves

    Zezhen Sun and Yuting WuWider flows

    Topics Volume-preserving & constrained flows, Other curvature flows · Shared terms “area preserving”, “plane curves”

  7. The asymptotic of the Mullins-Sekerka and the area-preserving curvature flow in the planar flat torus

    Vedansh Arya, Daniele De Gennaro and Anna KubinWider flows

    Topics Volume-preserving & constrained flows, Long-time existence & convergence, Other curvature flows · Shared terms “area preserving”, “asymptotic behavior”

  8. Asymptotic circularity of immortal area-preserving curvature flows

    Tatsuya MiuraWider flows

    Topics Volume-preserving & constrained flows, Long-time existence & convergence, Other curvature flows · Shared terms “area preserving”

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.

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) Volume-preserving & constrained flows Wider flows Curvature flows with a constraint: subtract the average of the mean curvature and the enclosed volume stays fixed while area decreases, so spheres… Start with: On an area-preserving evolution equation for plane curves (Gage, 1986) 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) Free boundary & capillary Flows whose boundary is free to slide on a support surface, meeting it at a right angle (free boundary) or at a fixed angle (capillary). Start with: Non-parametric mean curvature evolution with boundary conditions (Huisken, 1989)