Below the roughening temperature, a corrugated crystal surface develops facets at its peaks and valleys. The facets grow and merge, producing a uniformly flat
surface in finite time. A widely accepted PDE model for this process is “motion by surface diffusion” with a convex but non-smooth surface energy like
$\int|h_x| + |h_x|^3. This amounts to a highly nonlinear fourth-order parabolic PDE for the surface height h(x, t). I’ll discuss recent work with Irakli Odisharia, concerning:

• a robust numerical scheme for computing the evolution of h; and
• an explanation why the evolution is asymptotically self-similar.

The physical correctness of this PDE model remains uncertain. A natural approach would be to take the continuum limit of a step-flow model. I’ll discuss briefly our (incomplete) understanding of this limit.