  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 from the literature is shown to fail to produce the
correct asymptotics, albeit in a subtle manner. Two models are constructed that
approximate known sharp-interface equations without adding undesired
constraints. Linear stability of a planar interface is investigated for the
resulting phase-field equations and shown to reduce to the desired limit.
Finally, numerical simulations of the standard and a more sophisticated model
from the literature as well as of our two new models are performed to assess
the relative merits of each approach. The results suggest superior performance
of the new models in at least some situations.
