# American Institute of Mathematical Sciences

doi: 10.3934/dcdss.2021014
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## Energetics and coarsening analysis of a simplified non-linear surface growth model

 1 ENSA, LIPIM laboratory, University Sultan Moulay Slimane, Khouribga, Morocco 2 FST, LAMAI Laboratory, University Cadi Ayyad, Marrakesh, Morocco

* Corresponding author: Hamza Khalfi

Received  August 2020 Revised  December 2020 Early access January 2021

We study a simplified multidimensional version of the phenomenological surface growth continuum model derived in [6]. The considered model is a partial differential equation for the surface height profile
 $u$
which possesses the following free energy functional:
 $E(u) = \int_{\Omega} \left[ \frac{1}{2} \ln\left(1+\left|\nabla u \right|^2\right) - \left|\nabla u \right| \arctan\left(\left|\nabla u \right|\right) + \frac{1}{2} \left|\Delta u \right|^2 \right] {\rm d}x,$
where
 $\Omega$
is the domain of a fixed support on which the growth is carried out. The term
 $\left|\Delta u \right|^2$
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
 $\left( \left|\nabla u \right| \right)$
. 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
 $\left| \Omega \right|$
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.
Citation: Hamza Khalfi, Amal Aarab, Nour Eddine Alaa. Energetics and coarsening analysis of a simplified non-linear surface growth model. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021014
##### References:

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##### References:
Thin film surface height in a co-moving frame ($O{x_1}{x_2}z$)
Evolution dynamics of the front equation (10) starting from the initial condition (40) where $\varepsilon^2 = 0.25$, $\lambda = 10^{-2}$, $M_k = 2^8$ and $\gamma = 1$
Evolution of the energy (6) in one dimension starting from the initial configuration (40) where $\varepsilon = 0.25$, $\lambda = 10^{-2}$, $M_k = 2^8$ and $\gamma = 1$
Evolution of the energy (6) in two dimensions starting from the initial configuration (41) where $\varepsilon = 0.1$, $\lambda = 10^{-2}$, $M_k = 2^8$ and $\gamma = 0.2$
Evolution of the energy (6) in two dimensions starting from the initial configuration (41) where $\varepsilon = 0.01$, $\lambda = 10^{-2}$, $M_k = 2^8$ and $\gamma = 0.2$
Evolution of the interfacial width or roughness $W_u(t)$ starting from the initial configuration (41) where $\varepsilon = 0.1$, $\lambda = 10^{-2}$, $M_k = 2^8$ and $\gamma = 0.5$
Evolution of the $L^2$-error $\| u^{n+1} - u^{n} \|_{L^2}$ (6) starting from the initial configuration (41) where $\varepsilon = 0.01$, $\lambda = 10^{-2}$, $M_k = 2^8$ and $\gamma = 0.2$
Evolution dynamics of the front equation (10) starting from the initial condition (41) where $\varepsilon = 0.1$, $\lambda = 10^{-2}$, $M_k = 2^8$ and $\gamma = 0.2$
Evolution dynamics of the front equation (10) starting from the initial condition (41) where $\varepsilon = 0.01$, $\lambda = 10^{-2}$, $M_k = 2^8$ and $\gamma = 0.2$
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