The Role of Symmetry and Separation in Surface Evolution and Curve Shortening

Philip Broadbridge, Peter Vassiliou

    Research output: Contribution to journalArticle

    2 Citations (Scopus)

    Abstract

    With few exceptions, known explicit solutions of the curve shortening flow (CSE) of a plane curve, can be constructed by classical Lie point symmetry reductions or by functional separation of variables. One of the functionally separated solutions is the exact curve shortening flow of a closed, convex ''oval''-shaped curve and another is the smoothing of an initial periodic curve that is close to a square wave. The types of anisotropic evaporation coefficient are found for which the evaporation-condensation evolution does or does not have solutions that are analogous to the basic solutions of the CSE, namely the grim reaper travelling wave, the homothetic shrinking closed curve and the homothetically expanding grain boundary groove. Using equivalence classes of anisotropic diffusion equations, it is shown that physical models of evaporation-condensation must have a diffusivity function that decreases as the inverse square of large slope. Some exact separated solutions are constructed for physically consistent anisotropic diffusion equations
    Original languageEnglish
    Pages (from-to)1-19
    Number of pages19
    JournalSymmetry, Integrability and Geometry: Methods and Applications
    Volume7
    DOIs
    Publication statusPublished - 2011

    Fingerprint

    Symmetry
    Evaporation
    Curve
    Anisotropic Diffusion
    Condensation
    Diffusion equation
    Lie Point Symmetries
    Symmetry Reduction
    Closed curve
    Separation of Variables
    Grain Boundary
    Plane Curve
    Shrinking
    Diffusivity
    Physical Model
    Explicit Solution
    Equivalence class
    Traveling Wave
    Exception
    Smoothing

    Cite this

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    The Role of Symmetry and Separation in Surface Evolution and Curve Shortening. / Broadbridge, Philip; Vassiliou, Peter.

    In: Symmetry, Integrability and Geometry: Methods and Applications, Vol. 7, 2011, p. 1-19.

    Research output: Contribution to journalArticle

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