Anisotropic motion of a phase interface. Well-posedness of the initial value problem and qualitative properties of the interface.

*(English)*Zbl 0784.35124This paper continues the discussion of certain evolution equations for phase interfaces which were derived in an earlier paper [Arch. Ration. Mech. Anal. 108, No. 4, 323-391 (1989; Zbl 0723.73017)]. The most general model considered here deals with the evolution of a domain with piecewise smooth boundary. On the smooth parts the boundary moves according to a nondegenerate parabolic equation; at the vertices the model requires the capillary force to be continuous, which effectively means that at a corner point the tangent angles are prescribed.

Using the theory of analytic semigroups it is shown that the initial value problem under consideration has classical solutions on short enough time intervals, and that solutions can be continued until the phase interface develops self intersections, loses its local Lipschitz nature, or until one of the smooth segments of the interface vanishes.

In the second part of the paper various geometric properties of classical solutions are derived. For “admissible solutions” with convex corners a monotonicity principle is derived, saying that larger initial domains will have larger solutions. An example is presented showing that admissibility is a necessary assumption. The unique stationary shape is shown to be unstable, with a one-dimensional unstable manifold. By using this unstable manifold the asymptotic shape of unboundedly growing domains is found.

Using the theory of analytic semigroups it is shown that the initial value problem under consideration has classical solutions on short enough time intervals, and that solutions can be continued until the phase interface develops self intersections, loses its local Lipschitz nature, or until one of the smooth segments of the interface vanishes.

In the second part of the paper various geometric properties of classical solutions are derived. For “admissible solutions” with convex corners a monotonicity principle is derived, saying that larger initial domains will have larger solutions. An example is presented showing that admissibility is a necessary assumption. The unique stationary shape is shown to be unstable, with a one-dimensional unstable manifold. By using this unstable manifold the asymptotic shape of unboundedly growing domains is found.

Reviewer: S.B.Angenent (Madison)

##### MSC:

35R35 | Free boundary problems for PDEs |

35K65 | Degenerate parabolic equations |

74A15 | Thermodynamics in solid mechanics |

80A17 | Thermodynamics of continua |