Accepted Paper

Inserted: 19 apr 2017
Last Updated: 18 aug 2018

Journal: Commun. Math. Sci.
Year: 2017

Abstract:

We consider the evolution equation \begin{equation}\label{abs1} h_{t=\ddt \F^{-1}(-aE \F(h)) - r/h^2 -\ddt h ,} \end{equation} introduced in {\cite{TS}} by Tekalign and Spencer to describe the heteroepitaxial growth of a two-dimensional thin film on an elastic substrate. In the expression above, $h$ denotes the surface height of the film, $\F$ is the Fourier transform, and $a$, $E$, $r$ are positive material constants. For simplicity, we set $aE=r=1$. As this equation does not have any particular structure, its analysis is quite challenging. Therefore, we introduce the auxiliary equation (with $c$ being a given constant) \begin{equation}\label{abs2} u_{t=\gr - \div u - (\div u+c)^{-2} -\ddt \div u ,} \end{equation} which has a variational structure. Equivalency between \eqref{abs1} and \eqref{abs2} will hold under sufficient regularity on the solution. The main aim of this paper is to provide an analytical validation to \eqref{abs2}, by proving existence and regularity properties for weak solutions, under suitable assumptions on the initial datum.

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