Evolution of a Crystal Surface Below the Roughening Temperature
the roughening temperature, a corrugated crystal surface develops
facets at its peaks and valleys. The facets grow and merge, producing a
surface in finite time. A widely accepted PDE model for this process is
“motion by surface diffusion” with a convex but non-smooth surface
$\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.