On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation

Abstract

We present a formal asymptotic analysis which suggests a model for three-phase boundary motion as a singular limit of a vector-valued Ginzburg-Landau equation. We prove short-time existence and uniqueness of solutions for this model, that is, for a system of three-phase boundaries undergoing curvature motion with assigned angle conditions at the meeting point. Such models pertain to grain boundary motion in alloys. The method we use, based on linearization about the initial conditions, applies to a wide class of parabolic systems. We illustrate this further by its application to an eutectic solidification problem.

Nonlocal Allen-Cahn systems: analysis and a primal-dual active set method

“…Choosing χ ≡ ∂u δ /∂t in (12) and using P P P R ∂u δ ∂t = ∂u δ ∂t which is due to (13) and (14) we obtain…”

“…To demonstrate the efficiency of the method we also performed a computation with thirty order parameters. In this case the Allen-Cahn system models grain growth and at triple junctions a 2π 3 angle condition has to hold, see [13,24] for details. For the computation in Figure 3 we use a Voronoi partitioning algorithm to randomly fill the 2D computational domain.…”

“…It has been proved that for two regions the optimal configuration is a double bubble [28]. Due to the integral constraints the regions to be separated have fixed volume whilst the evolution tends to minimize the surface energy, and hence the surface area, see [13,23]. In the first computation we use three order parameters and start with a sphere where the left half is occupied by phase 1 and the right half is occupied by phase 2.…”

“…In many of these applications more than two phases occur. Therefore, the model has been extended to deal with N phases [13,24]. The phase field takes now the form of a vector-valued function u : Ω × (0, T ) → R N which describes the fractions of the phases, i.e.…”

Parametric approximation of surface clusters driven by isotropic and anisotropic surface energies

“…In the plane, existence of solutions for mean curvature flow and surface diffusion with triple junction points has been shown in [20] and [36], respectively. For the higher dimensional case it is known that very weak solutions exist for the mean curvature flow.…”