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.
We study entire solutions on R 2 of the elliptic system −∆U + ∇W (u) = 0 where W : R 2 → R 2 is a multiple-well potential. We seek solutions U (x 1 , x 2 ) which are "heteroclinic," in two senses: for each fixed x 2 ∈ R they connect (at x 1 = ±∞) a pair of constant global minima of W , and they connect a pair of distinct one dimensional stationary wave solutions when x 2 → ±∞. These solutions describe the local structure of solutions to a reaction-diffusion system near a smooth phase boundary curve. The existence of these heteroclinic solutions demonstrates an unexpected difference between the scalar and vector valued Allen-Cahn equations, namely that in the vectorial case the transition profiles may vary tangentially along the interface. We also consider entire stationary solutions with a "saddle" geometry, which describe the structure of solutions near a crossing point of smooth interfaces.
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