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Curvature Driven Interface Evolution

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Abstract

Curvature driven surface evolution plays an important role in geometry, applied mathematics and in the natural sciences. In this paper geometric evolution equations such as mean curvature flow and its fourth order analogue motion by surface diffusion are studied as examples of gradient flows of the area functional. Also in many free boundary problems the motion of an interface is given by an evolution law involving curvature quantities. We will introduce the Mullins-Sekerka flow and the Stefan problem with its anisotropic variants and discuss their properties.

In phase field models the area functional is replaced by a Ginzburg-Landau functional leading to a diffuse interface model. We derive the Allen-Cahn equation, the Cahn-Hilliard equation and the phase field system as gradient flows and relate them to sharp interface evolution laws.

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Notes

  1. Here we cite Ecker [54, p. 53].

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Acknowledgements

Figures 1–3, 5, 8–12 are numerical computations by Robert Nürnberg (Imperial College, London) and they were performed in the context of the work [8–13]. Figure 6 has been provided by Ulrich Weikard. Helmut Abels, Klaus Deckelnick, Daniel Depner, Hans-Christoph Grunau, Claudia Hecht, Barbara Niethammer and Matthias Röger made helpful suggestions which improved the presentation. I would like to express my gratitude to all the above mentioned colleagues for their contributions and to Eva Rütz for typing my often very rough notes.

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    Harald Garcke

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Garcke, H. Curvature Driven Interface Evolution. Jahresber. Dtsch. Math. Ver. 115, 63–100 (2013). https://doi.org/10.1365/s13291-013-0066-2

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