Regularity in Time for Weak Solutions of a Continuum Model for Epitaxial Growth with Elasticity on Vicinal Surfaces

TL;DR

This paper proves existence, uniqueness, and Lipschitz regularity in time for weak solutions of a continuum model describing epitaxial growth with elasticity on vicinal surfaces, addressing an open problem in the regularity of solutions.

Contribution

It establishes time regularity and well-posedness of weak solutions for a complex surface evolution PDE, extending previous existence results.

Findings

Existence of weak solutions confirmed.

Uniqueness of solutions established.

Solutions exhibit Lipschitz regularity in time.

Abstract

The evolution equation derived by Xiang (SIAM J. Appl. Math. 63:241--258, 2002) to describe vicinal surfaces in heteroepitaxial growth is $$ h_t=-\left[ H(h_x)+\left(h_x^{-1}+h_x \right) h_{xx}\right]_{xx}, $$ where $h$ denotes the surface height of the film, and $H$ is the Hilbert transform. Existence of solutions was obtained by Dal Maso, Fonseca and Leoni (Arch. Rational Mech. Anal. 212: 1037--1064, 2014). The regularity in time was left unresolved. The aim of this paper is to prove existence, uniqueness, and Lipschitz regularity in time for weak solutions, under suitable assumptions on the initial datum.