Self-similar solutions of a 2-D multiple-phase curvature flow

Abstract

Abstract. This article studies self-similar shrinking, stationary, and expanding solutions of a 2-dimensional motion by curvature equation modelling evolution of grain boundaries in polycrystals. Here the interfacial energy densities are assumed to depend only on the grains and the Herring condition is used for triple junctions (the intersections of three grain boundaries). In particular, in the isotropic case, a total of six configurations are classified as the only self-similar shrinking solutions.

Motion by curvature of planar curves with end points moving freely on a line

“…2 can be used. Here we follow our earlier paper [15] using polar coordinates. Hence, we seek solutions to (2.6) in the form…”

On the Classification of Networks Self–Similarly Moving by Curvature

“…In [3,4] we were able to complete the classi cation of the complete, embedded, self-similarly shrinking regular networks in the plane with at most two triple junctions, after the contributions in [7,9,20]. We describe such a classi cation here below.…”

“…It remains to discuss the last two cases: one is the "lens/ sh" shape and the other is the shape of the Greek "theta" letter (or "double cell"). It is well known that there exist unique (up to a rotation) lens-shaped and sh-shaped, complete, embedded, regular shrinkers, which are symmetric with respect to a line through the origin of R (see [7,20] shrinkers) exist, with numerical evidence in favor of the conjecture of non-existence (see [9]). We have proved that this is actually the case.…”

“…After the contributions in [7,9,20], such a classi cation was completed in the recent work in [4] (see also [3]), proving that there are no self-shrinking networks homeomorphic to the Greek "theta" letter (a double cell) embedded in the plane with two triple junctions with angles of degrees. We present the main geometric ideas behind the work [4].…”

On the mean curvature flow of grain boundaries

“…For three regular curves meeting at a triple junction of 120 degrees, Bronsard and Reitich [10] proved short-time existence and uniqueness using a theory of system of parabolic PDE [42]. There are numerous results studying existence, uniqueness (or non-uniqueness) and stability under various boundary conditions as well as studies on the self-similar shrinking/expanding solutions, and we mention [6,11,12,20,21,23,27,31,33,35,36,39,40]. Compared to the above known results, our existence theorem does not require any parametrization and there is no restriction on the dimension or configuration.…”