Paper · Mar 9, 2017

From conservative to dissipative systemsthrough quadratic change of time, with application to the curve-shortening flow

Yann Brenier, Xianglong Duan

math.APWider flowsarXiv:1703.03404
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Abstract

We provide several examples of dissipative systems that can be obtained from conservative ones through a simple, quadratic,change of time. A typical example is the curve-shortening flow in R^d, which is a particular case ofmean-curvature flow with co-dimension higher than one (except in the case d=2).Through such a change of time, this flow can be formally derived from the conservative model of vibrating strings obtainedfrom the Nambu-Goto action. Using the concept of "relative entropy" (or "modulated energy"), borrowed from the theoryof hyperbolic systems of conservation laws, we introduce a notion of generalized solutions,that we call dissipative solutions, for the curve-shortening flow. For given initial conditions, the set of generalized solutionsis convex, compact, if not empty. Smooth solutions to the curve-shortening flow are always unique in this setting.

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Submitted
Mar 9, 2017
Subjects
math.AP
DOI
10.1007/s00205-017-1166-4
Mean curvature flowCurve shortening

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 On the rate of convergence of the p-curve shortening flow (Jean C. Cortissoz, Andrés Galindo and Alexander Murcia, 2017)1Cortissoz 2017 Affine Self-similar solutions of the affine curve shortening flow I: The degenerate case (Chengjie Yu and Feifei Zhao, 2017)5Yu 2017 Ancient solutions to the Curve Shortening Flow spanning the halfplane (John Man Shun Ma, 2020)7Ma 2020 Self-similar curve shortening flow in hyperbolic 2-space (Eric Woolgar and Ran Xie, 2021)2Woolgar 2021 A Compact Non-convex Ancient Curve Shortening Flow (Jumageldi Charyyev, 2022)4Charyyev 2022 Non-uniqueness of curve shortening flow (Luke Thomas Peachey, 2022)3Peachey 2022 Compact curve shortening flow solutions out of non compact curve (Theodora Bourni and Martin Reiris, 2023)6Bourni 2023 On asymptotically conical curve shortening flows with convex projections (Alexander Mramor, 2026)8Mramor 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the rate of convergence of the -curve shortening flow

    Jean C. Cortissoz, Andrés Galindo and Alexander MurciaWider flows

    Topics Curve shortening

  2. Self-similar curve shortening flow in hyperbolic 2-space

    Eric Woolgar and Ran XieWider flows

    Topics Curve shortening

  3. Non-uniqueness of curve shortening flow

    Luke Thomas PeacheyWider flows

    Topics Curve shortening

  4. A Compact Non-convex Ancient Curve Shortening Flow

    Jumageldi CharyyevWider flows

    Topics Curve shortening

  5. Affine Self-similar solutions of the affine curve shortening flow I: The degenerate case

    Chengjie Yu and Feifei ZhaoWider flows

    Topics Curve shortening

  6. Compact curve shortening flow solutions out of non compact curve

    Theodora Bourni and Martin ReirisWider flows

    Topics Curve shortening

  7. Ancient solutions to the Curve Shortening Flow spanning the halfplane

    John Man Shun MaWider flows

    Topics Curve shortening

  8. On asymptotically conical curve shortening flows with convex projections

    Alexander MramorWider flows

    Topics Curve shortening

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) Curve shortening Wider flows Plane curves moving by their curvature. Convex curves shrink to round points (Gage–Hamilton), and every embedded curve becomes convex first (Grayson). Start with: The heat equation shrinking convex plane curves (Gage et al., 1986)

In the guidebook

  1. 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.

  2. 12C.4
    Loops, Disks and Curve ShrinkingBook 12C · Extinction and the Proof

    Perelman III §§2–3 and Morgan–Tian §§18.3–18.7: π₃ as spheres of loops, minimal disks and the −2π estimate, curve shrinking, ramps in M × S¹ and the estimates that make the argument rigorous.