# American Institute of Mathematical Sciences

## 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 Published  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:

show all references

##### 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 $\gamma = 0.2$
 [1] J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008 [2] Kazeem Olalekan Aremu, Chinedu Izuchukwu, Grace Nnenanya Ogwo, Oluwatosin Temitope Mewomo. Multi-step iterative algorithm for minimization and fixed point problems in p-uniformly convex metric spaces. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2161-2180. doi: 10.3934/jimo.2020063 [3] Liangliang Ma. Stability of hydrostatic equilibrium to the 2D fractional Boussinesq equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021068 [4] Haodong Chen, Hongchun Sun, Yiju Wang. A complementarity model and algorithm for direct multi-commodity flow supply chain network equilibrium problem. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2217-2242. doi: 10.3934/jimo.2020066 [5] Guodong Wang, Bijun Zuo. Energy equality for weak solutions to the 3D magnetohydrodynamic equations in a bounded domain. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021078 [6] De-han Chen, Daijun jiang. Convergence rates of Tikhonov regularization for recovering growth rates in a Lotka-Volterra competition model with diffusion. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021023 [7] Hongsong Feng, Shan Zhao. A multigrid based finite difference method for solving parabolic interface problem. Electronic Research Archive, , () : -. doi: 10.3934/era.2021031 [8] Nikolaz Gourmelon. Generation of homoclinic tangencies by $C^1$-perturbations. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 1-42. doi: 10.3934/dcds.2010.26.1 [9] Meng Ding, Ting-Zhu Huang, Xi-Le Zhao, Michael K. Ng, Tian-Hui Ma. Tensor train rank minimization with nonlocal self-similarity for tensor completion. Inverse Problems & Imaging, 2021, 15 (3) : 475-498. doi: 10.3934/ipi.2021001 [10] Kha Van Huynh, Barbara Kaltenbacher. Some application examples of minimization based formulations of inverse problems and their regularization. Inverse Problems & Imaging, 2021, 15 (3) : 415-443. doi: 10.3934/ipi.2020074 [11] Junichi Minagawa. On the uniqueness of Nash equilibrium in strategic-form games. Journal of Dynamics & Games, 2020, 7 (2) : 97-104. doi: 10.3934/jdg.2020006 [12] Christopher Bose, Rua Murray. Minimum 'energy' approximations of invariant measures for nonsingular transformations. Discrete & Continuous Dynamical Systems, 2006, 14 (3) : 597-615. doi: 10.3934/dcds.2006.14.597 [13] Monica Conti, Lorenzo Liverani, Vittorino Pata. A note on the energy transfer in coupled differential systems. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021042 [14] Didier Bresch, Thierry Colin, Emmanuel Grenier, Benjamin Ribba, Olivier Saut. A viscoelastic model for avascular tumor growth. Conference Publications, 2009, 2009 (Special) : 101-108. doi: 10.3934/proc.2009.2009.101 [15] Arunima Bhattacharya, Micah Warren. $C^{2, \alpha}$ estimates for solutions to almost Linear elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021024 [16] Amanda E. Diegel. A C0 interior penalty method for the Cahn-Hilliard equation. Electronic Research Archive, , () : -. doi: 10.3934/era.2021030 [17] Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196 [18] Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022 [19] Suzete Maria Afonso, Vanessa Ramos, Jaqueline Siqueira. Equilibrium states for non-uniformly hyperbolic systems: Statistical properties and analyticity. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021045 [20] Filippo Giuliani. Transfers of energy through fast diffusion channels in some resonant PDEs on the circle. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021068

2019 Impact Factor: 1.233

## Tools

Article outline

Figures and Tables