The evolution of a crystal surface: Analysis of a one-dimensional step train connecting two facets in the ADL regime

We study the evolution of a monotone step train separating two facets of a crystal surface. The model is one-dimensional and we consider only the attachment–detachment-limited regime. Starting with the well-known ODEs for the velocities of the steps, we consider the system of ODEs giving the evolution of the “discrete slopes.” It is the l 2 -steepest-descent of a certain functional. Using this structure, 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, identifying its regularity, and discussing its qualitative behavior. Our approach suggests a PDE for the slope as a function of height and time in the continuum setting. However, existence, uniqueness, and asymptotic self-similarity remain open for the continuum version of the problem.