Energetics and coarsening analysis of a simplified non-linear surface growth model

Abstract

<p style='text-indent:20px;'>We study a simplified multidimensional version of the phenomenological surface growth continuum model derived in [<xref ref-type="bibr" rid="b6">6</xref>]. The considered model is a partial differential equation for the surface height profile <inline-formula><tex-math id="M1">\begin{document}$ u $\end{document}</tex-math></inline-formula> which possesses the following free energy functional:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ E(u) = \int_{\Omega} \left[ \frac{1}{2} \ln\left(1+\left|\nabla u \right|^2\right) - \left|\nabla u \right| \arctan\left(\left|\nabla u \right|\right) + \frac{1}{2} \left|\Delta u \right|^2 \right] {\rm d}x, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M2">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is the domain of a fixed support on which the growth is carried out. The term <inline-formula><tex-math id="M3">\begin{document}$ \left|\Delta u \right|^2 $\end{document}</tex-math></inline-formula> designates the standard surface diffusion in contrast to the second order term which phenomenologically describes the growth instability. The energy above is mainly carried out in regions of higher surface slope <inline-formula><tex-math id="M4">\begin{document}$ \left( \left|\nabla u \right| \right) $\end{document}</tex-math></inline-formula>. Hence minimizing such energy corresponds to reducing surface defects during the growth process from a given initial surface configuration. Our analysis is concerned with the energetic and coarsening behaviours of the equilibrium solution. The existence of global energy minimizers and a scaling argument are used to construct a sequence of equilibrium solutions with different wavelength. We apply our minimum energy estimates to derive bounds in terms of the linear system size <inline-formula><tex-math id="M5">\begin{document}$ \left| \Omega \right| $\end{document}</tex-math></inline-formula> for the characteristic interface width and average slope. We also derive a stable numerical scheme based on the convex-concave decomposition of the energy functional and study its properties while accommodating these results with 1d and 2d numerical simulations.</p>