Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 052, 19 pages      arXiv:1106.0092
Contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S4)”

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

Philip Broadbridge a and Peter Vassiliou b
a) School of Engineering and Mathematical Sciences, La Trobe University, Melbourne, Victoria, Australia
b) Faculty of Information Sciences and Engineering, University of Canberra, Canberra, A.C.T., Australia

Received January 23, 2011, in final form May 25, 2011; Published online June 01, 2011

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.

Key words: curve shortening flow; exact solutions; symmetry; separation of variables.

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  1. Abresch U., Langer J. The normalized curve shortening flow and homothetic solutions, J. Differential Geom. 23 (1986), 175-196.
  2. Angenent S., On the formation of singularities in the curve shortening flow, J. Differential Geom. 33 (1991), 601-633.
  3. Arrigo D.J., Broadbridge P., Tritscher P., Karciga Y., The depth of a steep evaporating grain boundary groove: application of comparison theorems, Math. Comput. Modelling 25 (1997), no. 10, 1-8.
  4. Bluman G.W., Reid G.J., Kumei S., New classes of symmetries for partial differential equations, J. Math. Phys. 29 (1988), 806-811, Erratum, J. Math. Phys. 29 (1988), 2320.
  5. Broadbridge P., Exact solvability of the Mullins nonlinear diffusion model of groove development, J. Math. Phys. 30 (1989), 1648-1651.
  6. Broadbridge P., Tritscher P., An integrable fourth-order nonlinear evolution equation applied to thermal grooving of metal surfaces, IMA J. Appl. Math. 53 (1994), 249-265.
  7. Broadbridge P., Goard J.M., Temperature-dependent surface diffusion near a grain boundary, J. Engrg. Math. 66 (2010), 87-102.
  8. Cahn J.W., Taylor J.E., Overview no. 113 surface motion by surface diffusion, Acta Metall. Mater. 42 (1994), 1045-1063.
  9. Cao F., Geometric curve evolution and image processing, Lecture Notes in Mathematics, Vol. 1805, Springer-Verlag, Berlin, 2003.
  10. Carslaw H.S., Jaeger J.C. Conduction of heat in solids, 2nd. ed., Clarendon Press, Oxford, 1959.
  11. Chou K.-S., Li G.-X., Optimal systems and invariant solutions for the curve shortening problem, Comm. Anal. Geom. 10 (2002), 241-274.
  12. Clarkson P.A., Fokas A.S., Ablowitz M.J., Hodograph transformations of linearizable partial differential equations, SIAM J. Appl. Math. 49 (1989), 1188-1209.
  13. Daskalopoulos P., Hamilton R., Sesum N., Classification of compact ancient solutions to the curve shortening flow, J. Differential Geom. 84 (2010), 455-465, arXiv:0806.1757.
  14. Doyle P.W., Vassiliou P.J., Separation of variables for the 1-dimensional non-linear diffusion equation, Internat. J. Non-Linear Mech. 33 (1998), 315-326.
  15. Galaktionov V.A., Dorodnitsyn V.A., Elenin G.G., Kurdyumov S.P., Samarskii A.A., A quasilinear equation of heat conduction with a source: peaking, localization, symmetry, exact solutions, asymptotic behavior, structures, J. Soviet Math. 41 (1988), 1222-1292.
  16. Gage M., Hamilton R., The heat equation shrinking convex plane curves, J. Differential Geom. 23 (1986), 69-96.
  17. Grayson M. 1987 The heat equation shrinks embedded plane curves to round points, J. Differential Geom. 26 (1987), 285-314.
  18. Halldorsson H.P., Self-similar solutions to the curve shortening flow, arXiv:1007.1617.
  19. Herring C., Surface tension as a motivation for sintering, in The Physics of Powder Metallurgy, Editor W.E. Kingston, McGraw-Hill, 1951, 143-179.
  20. Ishimura N., Curvature evolution of plane curves with prescribed opening angle, Bull. Austral. Math. Soc. 52 (1995), 287-296.
  21. King J.R., Emerging areas of mathematical modelling, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 358 (2000), no. 1765, 3-19.
  22. Kingston J.G., Rogers C., Reciprocal Bäcklund transformations of conservation laws, Phys.  Lett. A 92 (1982), 261-264.
  23. Malladi R., Sethian J.A., Image processing via level set curvature flow, Proc. Nat. Acad. Sci. USA 92 (1995), 7046-7050.
  24. Mullins W.W., Theory of thermal grooving, J. Appl. Phys. 28 (1957), 333-339.
  25. Olver P.J., Sapiro G., Tannenbaum A., Invariant geometric evolutions of surfaces and volumetric smoothing, SIAM J. Appl. Math. 57 (1997), 176-194.
  26. Tritscher P., Integrable nonlinear evolution equations applied to solidification and surface redistribution, PhD Thesis, University of Wollongong, 1995.
  27. Tritscher P., An integrable fourth-order nonlinear evolution equation applied to surface redistribution due to capillarity, J. Austral. Math. Soc. Ser. B 38 (1997), 518-541.
  28. Tritscher P., Broadbridge P., Grain boundary grooving by surface diffusion: an analytic nonlinear model for a symmetric groove, Proc. Roy. Soc. Lond. A 450 (1995), no. 1940, 569-587.

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