# Analysis of a continuum theory for broken bond crystal surface models with evaporation and deposition effects

## TL;DR

This paper analyzes a modified fourth-order PDE model for crystal surface evolution with evaporation and deposition, providing a derivation, existence proof of weak solutions, and discussing smooth solutions for small data.

## Contribution

It offers a non-rigorous derivation and a rigorous proof of global weak solutions for a continuum model of crystal surfaces, extending methods to similar degenerate fourth-order equations.

## Key findings

- Existence of global weak solutions established.
- Derivation of the PDE model from interaction potentials.
- Discussion of smooth solutions for small initial data.

## Abstract

We study a $4$th order PDE with a $2$nd order correction that is a modification of one suggested by Peter Smereka using the tools of Krug et al (Z. Phys. B., 1994). First, we provide a non-rigorous derivation of the full model derived with a quadratic interaction potential using the rates and methods of the last author with Jonathan Weare (PRE, 2013). Then, we prove the existence of a global in time weak solution by regularizing the equation in a way that allows us to apply the tools of Bernis-Friedman (JDE, 1990). The methods developed here can be applied to a large number of $4$th order degenerate 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 (IFB, 2018).