Gradient bounds for a thin film epitaxy equation

TL;DR

This paper analyzes a gradient flow model for thin film epitaxy, establishing well-posedness and bounds on the surface slope to understand slope selection mechanisms.

Contribution

It provides the first optimal local and global well-posedness results for the nonlinear diffusion equation with biharmonic dissipation in this context.

Findings

Established optimal local and global well-posedness for the model.

Derived bounds for the gradient of solutions in dimensions up to 3.

Provided insights into the slope selection mechanism and dissipation effects.

Abstract

We consider a gradient flow modeling the epitaxial growth of thin films with slope selection. The surface height profile satisfies a nonlinear diffusion equation with biharmonic dissipation. We establish optimal local and global wellposedness for initial data with critical regularity. To understand the mechanism of slope selection and the dependence on the dissipation coefficient, we exhibit several lower and upper bounds for the gradient of the solution in physical dimensions $d\le 3$.