The motion of such interfaces during gradient flow in the norm reduces to the same motion as surface diffusion, if the mobility in equation 13 is independent of C within the interface where the gradient of C is large but decreases to a negligible amount in the bulk phases. In that case, equation 10 for gradient flow in the norm results in level sets of C within the interface that move with approximate normal velocity:
The correspondence with surface diffusion is exact in the asymptotic limit of sharp interfaces, even when M depends interface orientation. This is the equation of volume conserving motion of an anisotropic surface by surface diffusion, and in the isotropic case reduces to Mullins' equation of motion by (minus) the Laplacian of the geometric mean curvature. Equation 15 is obtained from gradient flow on the surface integral , where is the anisotropic surface energy. Sharp and diffuse interface motion equations in gradient flow give equivalent expressions for volume conserving interface motion.
The motion such interfaces during gradient flow in the norm is appropriate for grain boundaries, domain walls, or any other boundary for which long-range diffusion is not required for conservation. In the limit of sharp interfaces, motion of the level sets during gradient flow is the same as that which is derived from the surface integral approach in table III:
which holds whether or not the interface is anisotropic.
The variational principle which leads to this motion is used in the next section to calculate the evolution of triple junctions.