Jahresbericht der Deutschen Mathematiker-Vereinigung

, Volume 115, Issue 2, pp 63–100

# Curvature Driven Interface Evolution

Survey Article

## 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.

### Keywords

Mean curvature flow Gradient flow Surface diffusion Mullins-Sekerka problem Stefan problem Crystal growth Phase field equation Allen-Cahn equation Cahn-Hilliard equation

### Mathematics Subject Classification (2000)

53C44 53K93 35K91 35R35 35K55 49Q20 53A10 80A22 82B24

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