On a Higher Order Convective Cahn--Hilliard-Type Equation

Abstract

A higher order convective Cahn--Hilliard-type equation that describes the faceting of a growing surface is considered with periodic boundary conditions. By using a Galerkin approach, the existence of weak solutions to this sixth order partial differential equation is established in $L^2(0,T;\dot{H}^3_{per})$. Additionally, stronger regularity results are derived. These are used to prove uniqueness of the solutions. Furthermore a numerical study shows that the transition from coarsening to roughening leads to multidomain-wall stationary and traveling states in a small deposition rate regime and to chaotic surfaces for increased rates. A characteristic length scale decreases logarithmically with increasing deposition rate.