One - sided Mullins - Sekerka Flow Does Not Preserve Convexity ∗ Uwe

The Mullins-Sekerka model is a nonlocal evolution model for hypersurfaces, which arises as a singular limit for the Cahn-Hilliard equation. Assuming the existence of sufficiently smooth solutions we will show that the one-sided Mullins-Sekerka flow does not preserve convexity. Introduction The Mullins-Sekerka flow is a nonlocal generalization of the mean curvature flow arising from physics [10, 11]. Similar to Stefan-type problems there is a one-sided and a two-sided version. Recently it has been shown rigorously that the two-sided model arises as a singular limit of the Cahn-Hilliard equation [1]. This has been known formally since the work of Pego [11]. In the literature the Mullins-Sekerka model has been often called Hele-Shaw model. However, there are two different problems which are called Hele-Shaw problems, compare for example [1, 2] with [4]. The problem studied in this paper is the same as the one-sided version of the Hele-Shaw problem as formulated in [1, 2]. To avoid this confusion one should probably call the Hele-Shaw flow of [1, 2] the Mullins-Sekerka flow. One can ask whether the properties of the mean curvature flow can be generalized to the Mullins-Sekerka flow. Not all results can be expected to generalize, due to the nonlocal character of the Mullins-Sekerka problem, in particular not those that rest on a local argument for the mean curvature flow. There has been some progress made towards the question of existence, see [5] for the one-sided version and [2] for the two-sided version. Recently Luckhaus has announced further results concerning existence, however, no details are know by the author. It is known that the mean curvature flow preserves convexity [6, 9]. It is therefore a natural question to ask whether this is also true for the Mullins-Sekerka flow. ∗1991 Mathematics Subject Classifications: 35R35, 35J05, 35B50, 53A07.