Analysis of a continuum theory for broken bond crystal surface models with evaporation and deposition effects**Dedicated to Peter Smereka (1959–2015) for his inspiration and insights.

Abstract

We study a 4th order degenerate parabolic PDE model in one-dimension with a 2nd order correction modeling the evolution of a crystal surface under the influence of both thermal fluctuations and evaporation/deposition effects. First, we provide a non-rigorous derivation of the PDE from an atomistic model using variations on kinetic Monte Carlo rates proposed by the last author with Weare [Marzuola J L and Weare J 2013 Phys. Rev. E 88 032403]. Then, we prove the existence of a global in time weak solution for the PDE by regularizing the equation in a way that allows us to apply the tools of Bernis–Friedman [Bernis F and Friedman A 1990 J. Differ. Equ. 83 179–206]. The methods developed here can be applied to a large number of 4th order degenerate PDE models. In an appendix, we also discuss the global smooth solution with small data in the Weiner algebra framework following recent developments using tools of the second author with Robert Strain [Liu J G and Strain R M 2019 Interfaces Free Boundaries 21 51–86].