,
can give spinodal decomposition and order-disorder kinetics as
equations such as
12, 13, and 14.
The solutions to these equations
describe the formation of interfaces and their
motion for small 
.
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[10].
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.
