Two-sided Mullins-Sekerka flow does not preserve convexity

Uwe F. Mayer

Abstract: The (two-sided) Mullins-Sekerka model is a nonlocal evolution model for closed hypersurfaces, which was originally proposed as a model for phase transitions of materials of negligible specific heat. Under this evolution the propagating interfaces maintain the enclosed volume while the area of the interfaces decreases. We will show by means of an example that the Mullins-Sekerka flow does not preserve convexity in two space dimensions, where we consider both the Mullins-Sekerka model on a bounded domain, and the Mullins-Sekerka model defined on the whole plane.

Key words: Mullins-Sekerka flow, Hele-Shaw flow, Cahn-Hilliard equation, maximum principle, free boundary problem, convexity, curvature.

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