Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 344 / 2 / e / Solution

Using only this upper-phase flux in $\dot{h}=v_n=-J_n/(2{\varphi }_B)$ givesPositive surface tension raises the chemical potential at a crest, drives material away from it, and therefore smooths the interface, which fixes the negative sign of the decay rate.

The dynamics is nonlocal because the conserved composition must diffuse through the bulk. Solving Laplace's equation maps a boundary Fourier amplitude to its normal derivative by multiplication by $|q|$. Combining this Dirichlet-to-Neumann factor with the curvature factor $q^2$ produces the nonanalytic $|q{|}^3$ rate described by diffusion-limited relaxation of an interface.