Grain boundary motion arising from the gradient flow of the Aviles-Giga functional

This paper considers the singular limit of the equation Θt=-εΔ2Θ+ε-1∇·([|∇Θ|2-1] ∇Θ). Grain boundaries (limiting discontinuities in ∇Θ) form networks that coarsen over time. A matched asymptotic analysis is used to derive a free boundary problem consisting of curve motion coupled along hyperbolic characteristics and junction conditions. An intermediate boundary layer near extrema junctions is discovered, along with the relevant nonlocal junction conditions. The limiting dynamics can be viewed in the context of a gradient flow of the sharp interface energy on an attracting manifold. Dynamic scaling of the long-time coarsening process can be explained by dimensional analysis of the reduced problem.