Abstract:
I'll discuss joint work with Hala Al Hajj Shehadeh and Jonathan Weare, concerning
the evolution of a monotone step train separating two facets of a crystal surface in the attachment-detachment-limited regime. Starting with the well-known ODE's for the velocities of the steps, we consider the system of ODE's giving the evolution of the ``discrete slopes.'' It has a natural gradient-flow structure. Using this, we prove that the solution exists for all time and is asymptotically self-similar. We also discuss the continuum limit of the discrete self-similar solution, characterizing it variationally and discussing its qualitative behavior. Our approach indicates a PDE for the slope as a function of height and time in the continuum setting. However existence, uniqueness, and asymptotic self-similarity remain unproved for the continuum version of the problem. |