# Regularity and monotonicity for solutions to a continuum model of epitaxial growth with nonlocal elastic effects

Journal Article (Journal Article)

We study the following parabolic nonlocal 4-th order degenerate equation: Ut=-[2â πâ Hâ (ux)+lnâi (uxâ x+a)+32â (uxâ x+a)2]xâ x,u{t}=-\Bigl{[}2\pi H(u{x})+\ln(u{XX}+a)+\frac{3}{2}(u{XX}+a){2}\Bigr{]}{% XX}, arising from the epitaxial growth on crystalline materials. Here H denotes the Hilbert transform, and a>0{a>0} is a given parameter. By relying on the theory of gradient flows, we first prove the global existence of a variational inequality solution with a general initial datum. Furthermore, to obtain a global strong solution, the main difficulty is the singularity of the logarithmic term when uxâ x+a{u{XX}+a} approaches zero. Thus we show that, if the initial datum u0{u{0}} is such that (u0)xâ x+a{(u{0}){XX}+a} is uniformly bounded away from zero, then such property is preserved for all positive times. Finally, we will prove several higher regularity results for this global strong solution. These finer properties provide a rigorous justification for the global-in-time monotone solution to the epitaxial growth model with nonlocal elastic effects on vicinal surface.

### Full Text

### Duke Authors

### Cited Authors

- Gao, Y; Lu, XY; Wang, C

### Published Date

- January 1, 2021

### Published In

### Electronic International Standard Serial Number (EISSN)

- 1864-8266

### International Standard Serial Number (ISSN)

- 1864-8258

### Digital Object Identifier (DOI)

- 10.1515/acv-2020-0114

### Citation Source

- Scopus