  It is demonstrated that the description of surface-diffusion controlled
dynamics via the phase-field method is less trivial than it appears at first
sight. A seemingly straightforward approach previously used in the literature
is shown to fail to produce the correct asymptotics, albeit in a subtle manner.
An apparently obvious alternative fails for a complementary reason. Finally, a
model is constructed that asymptotically approaches known sharp-interface
equations without adding undesired constraints. In order to provide a complete
physical example, the model is exhibited for the elastically induced Grinfeld
instability with material transport by surface diffusion. The whole analysis is
carried out for three-dimensional systems to pave the way for simulations in
3D.
