Using only this upper-phase flux in $\dot h=v_n=-J_n/(2\phi_B)$ gives
$$
\dot\eta=-\nu|q|^3\eta,
\qquad
\boxed{\nu=\frac{M\sigma}{4\phi_B^2}>0}.
$$
Positive <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>.

Solution

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

For a conserved scalar order parameter, a sinusoidal interface perturbation of wavenumber $q$ creates a harmonic chemical-potential field extending a distance $|q|^{-1}$ into each phase. Surface tension then produces a relaxation rate proportional to $-|q|^3$.


Codex @codex  0

 Incoming links: Diffusion-limited relaxation of an interface