American Institute of Mathematical Sciences

Advanced
December  2018, 11(6): 1359-1376. doi: 10.3934/krm.2018053

On the convergence to critical scaling profiles in submonolayer deposition models

Fernando P. da Costa 1,2,, , João T. Pinto 2,3, and  Rafael Sasportes 1,2,

1. 

Departamento de Ciȇncias e Tecnologia, Universidade Aberta, Lisboa, Portugal
2. 

Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal
3. 

Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal

* Corresponding author: F.P. da Costa

Received  July 2017 Published  June 2018

Fund Project: Partially funded by FCT/Portugal through project RD0447/CAMGSD/2015.

In this work we study the rate of convergence to similarity profiles in a mean field model for the deposition of a submonolayer of atoms in a crystal facet, when there is a critical minimal size $ n≥ 2$ for the stability of the formed clusters. The work complements recently published related results by the same authors in which the rate of convergence was studied outside of a critical direction $ x = τ$ in the cluster size $ x$ vs. time $ \tau $ plane. In this paper we consider a different similarity variable, $ ξ : = (x-\tau )/\sqrt \tau $, corresponding to an inner expansion of that critical direction, and prove the convergence of solutions to a similarity profile $ Φ_{2,n}(ξ)$ when $ x, \tau \to +∞$ with $ ξ$ fixed, as well as the rate at which the limit is approached.

Keywords: Dynamics of ODEs, coagulation processes, convergence to scaling behaviour, asymptotic evaluation of integrals, submonolayer deposition model.
Mathematics Subject Classification: Primary: 34C11, 34C20, 34C45, 34D05; Secondary: 82C21.
Citation: Fernando P. da Costa, João T. Pinto, Rafael Sasportes. On the convergence to critical scaling profiles in submonolayer deposition models. Kinetic & Related Models, 2018, 11 (6) : 1359-1376. doi: 10.3934/krm.2018053
References:
[1]

J. M. Ball, J. Carr and O. Penrose, The Becker-Döring cluster equations: Basic properties and asymptotic behaviour of solutions, Comm. Math. Phys., 104 (1986), 657-692.  doi: 10.1007/BF01211070.  Google Scholar

[2]

J. Cañizo, A. Einav and B. Lods, Trend to equilibrium for the Becker-Döring equation: an analogue of Cercignani's conjecture, Analysis and PDE, 10 (2017), 1663-1708.  doi: 10.2140/apde.2017.10.1663.  Google Scholar

[3]

J. Cañizo and B. Lods, Exponential convergence to equilibrium for subcritical solutions of the Becker-Döring equations, J. Diff. Equa., 225 (2013), 905-950.  doi: 10.1016/j.jde.2013.04.031.  Google Scholar

[4]

J. Cañizo, S. Mischler and C. Mouhot, Rate of convergence to self-similarity for Smoluchowski's coagulation equation with constant coefficients, SIAM J. Math. Anal., 41 (2010), 2283-2314.  doi: 10.1137/08074091X.  Google Scholar

[5]

F. P. da Costa, Mathematical Aspects of Coagulation-Fragmentation Equations, in Mathematics of Energy and Climate Change (eds. J. P. Bourguignon, R. Jeltsch, A. Pinto and M. Viana), CIM Series in Mathematical Sciences, vol. 2, Springer, Cham, (2015), 83–162.  Google Scholar

[6]

F. P. da Costa, J. T. Pinto and R. Sasportes, Rates of convergence to scaling profiles in a submonolayer deposition model and the preservation of memory of the initial condition, SIAM J. Math. Anal., 48 (2016), 1109-1127.  doi: 10.1137/15M1035033.  Google Scholar

[7]

F. P. da Costa, H. van Roessel and J. A. D. Wattis, Long-time behaviour and self-similarity in a coagulation equation with input of monomers, Markov Processes Relat. Fields, 12 (2006), 367-398.   Google Scholar

[8]

O. Costin, M. Grinfeld, K. P. O'Neill and H. Park, Long-time behaviour of point islands under fixed rate deposition, Commun. Inf. Syst., 13 (2013), 183-200.  doi: 10.4310/CIS.2013.v13.n2.a3.  Google Scholar

[9]

M. Einax, W. Dieterich and Ph. Maass, Colloquim: Cluster growth on surfaces: Densities, size distributions, and morphologies, Rev. Modern Phys., 85 (2013), 921-939.   Google Scholar

[10]

J. W. Evans and M. C. Bartelt, Nucleation, growth, and kinetic roughening of metal (100) homoepitaxial thin films, Langmuir, 12 (1996), 217-229.  doi: 10.1021/la940698s.  Google Scholar

[11]

P.-E. Jabin and B. Niethammer, On the rate of convergence to equilibrium in the Becker-Döring equations, J. Diff. Equa., 191 (2003), 518-543.  doi: 10.1016/S0022-0396(03)00021-4.  Google Scholar

[12]

Ph. Laurençot and S. Mischler, From the Becker-Döring to the Lifshitz-Slyozov-Wagner equations, J. Stat. Phys., 106 (2002), 957-991.  doi: 10.1023/A:1014081619064.  Google Scholar

[13]

Ph. Laurençot and S. Mischler, Liapunov functional for Smoluchowski's coagulation equation and convergence to self-similarity, Monatsh. Math., 146 (2005), 127-142.  doi: 10.1007/s00605-005-0308-1.  Google Scholar

[14]

G. Menon and R. L. Pego, Approach to self-similarity in Smoluchowski's coagulation equations, Commun. Pure Appl. Math., 57 (2004), 1197-1232.  doi: 10.1002/cpa.3048.  Google Scholar

[15]

P. A. Mulheran, Theory of cluster growth on surfaces, in Metallic Nanoparticles, vol. 5 of Handbook of Metal Physics (ed. J. A. Blackman), (series editor: P. Misra), Elsevier, Amsterdam, (2009), 73–111. Google Scholar

[16]

R. W. Murray and R. L. Pego, Algebraic decay to equilibrium for the Becker-Döring equations, SIAM J. Math. Anal., 48 (2016), 2819-2842.  doi: 10.1137/15M1038578.  Google Scholar

[17]

B. Niethammer, On the evolution of large clusters in the Becker-Döring model, J. Nonlinear Sci., 13 (2003), 115-155.  doi: 10.1007/s00332-002-0535-8.  Google Scholar

[18]

R. Srinivasan, Rates of convergence for Smoluchowski's coagulation equations, SIAM J. Math. Anal., 43 (2011), 1835-1854.  doi: 10.1137/090759707.  Google Scholar

[19]

J. A. D. Wattis, An introduction to mathematical models of coagulation-fragmentation processes: a discrete deterministic mean-field approach, Physica D, 222 (2006), 1-20.  doi: 10.1016/j.physd.2006.07.024.  Google Scholar

show all references

References:
[1]

J. M. Ball, J. Carr and O. Penrose, The Becker-Döring cluster equations: Basic properties and asymptotic behaviour of solutions, Comm. Math. Phys., 104 (1986), 657-692.  doi: 10.1007/BF01211070.  Google Scholar

[2]

J. Cañizo, A. Einav and B. Lods, Trend to equilibrium for the Becker-Döring equation: an analogue of Cercignani's conjecture, Analysis and PDE, 10 (2017), 1663-1708.  doi: 10.2140/apde.2017.10.1663.  Google Scholar

[3]

J. Cañizo and B. Lods, Exponential convergence to equilibrium for subcritical solutions of the Becker-Döring equations, J. Diff. Equa., 225 (2013), 905-950.  doi: 10.1016/j.jde.2013.04.031.  Google Scholar

[4]

J. Cañizo, S. Mischler and C. Mouhot, Rate of convergence to self-similarity for Smoluchowski's coagulation equation with constant coefficients, SIAM J. Math. Anal., 41 (2010), 2283-2314.  doi: 10.1137/08074091X.  Google Scholar

[5]

F. P. da Costa, Mathematical Aspects of Coagulation-Fragmentation Equations, in Mathematics of Energy and Climate Change (eds. J. P. Bourguignon, R. Jeltsch, A. Pinto and M. Viana), CIM Series in Mathematical Sciences, vol. 2, Springer, Cham, (2015), 83–162.  Google Scholar

[6]

F. P. da Costa, J. T. Pinto and R. Sasportes, Rates of convergence to scaling profiles in a submonolayer deposition model and the preservation of memory of the initial condition, SIAM J. Math. Anal., 48 (2016), 1109-1127.  doi: 10.1137/15M1035033.  Google Scholar

[7]

F. P. da Costa, H. van Roessel and J. A. D. Wattis, Long-time behaviour and self-similarity in a coagulation equation with input of monomers, Markov Processes Relat. Fields, 12 (2006), 367-398.   Google Scholar

[8]

O. Costin, M. Grinfeld, K. P. O'Neill and H. Park, Long-time behaviour of point islands under fixed rate deposition, Commun. Inf. Syst., 13 (2013), 183-200.  doi: 10.4310/CIS.2013.v13.n2.a3.  Google Scholar

[9]

M. Einax, W. Dieterich and Ph. Maass, Colloquim: Cluster growth on surfaces: Densities, size distributions, and morphologies, Rev. Modern Phys., 85 (2013), 921-939.   Google Scholar

[10]

J. W. Evans and M. C. Bartelt, Nucleation, growth, and kinetic roughening of metal (100) homoepitaxial thin films, Langmuir, 12 (1996), 217-229.  doi: 10.1021/la940698s.  Google Scholar

[11]

P.-E. Jabin and B. Niethammer, On the rate of convergence to equilibrium in the Becker-Döring equations, J. Diff. Equa., 191 (2003), 518-543.  doi: 10.1016/S0022-0396(03)00021-4.  Google Scholar

[12]

Ph. Laurençot and S. Mischler, From the Becker-Döring to the Lifshitz-Slyozov-Wagner equations, J. Stat. Phys., 106 (2002), 957-991.  doi: 10.1023/A:1014081619064.  Google Scholar

[13]

Ph. Laurençot and S. Mischler, Liapunov functional for Smoluchowski's coagulation equation and convergence to self-similarity, Monatsh. Math., 146 (2005), 127-142.  doi: 10.1007/s00605-005-0308-1.  Google Scholar

[14]

G. Menon and R. L. Pego, Approach to self-similarity in Smoluchowski's coagulation equations, Commun. Pure Appl. Math., 57 (2004), 1197-1232.  doi: 10.1002/cpa.3048.  Google Scholar

[15]

P. A. Mulheran, Theory of cluster growth on surfaces, in Metallic Nanoparticles, vol. 5 of Handbook of Metal Physics (ed. J. A. Blackman), (series editor: P. Misra), Elsevier, Amsterdam, (2009), 73–111. Google Scholar

[16]

R. W. Murray and R. L. Pego, Algebraic decay to equilibrium for the Becker-Döring equations, SIAM J. Math. Anal., 48 (2016), 2819-2842.  doi: 10.1137/15M1038578.  Google Scholar

[17]

B. Niethammer, On the evolution of large clusters in the Becker-Döring model, J. Nonlinear Sci., 13 (2003), 115-155.  doi: 10.1007/s00332-002-0535-8.  Google Scholar

[18]

R. Srinivasan, Rates of convergence for Smoluchowski's coagulation equations, SIAM J. Math. Anal., 43 (2011), 1835-1854.  doi: 10.1137/090759707.  Google Scholar

[19]

J. A. D. Wattis, An introduction to mathematical models of coagulation-fragmentation processes: a discrete deterministic mean-field approach, Physica D, 222 (2006), 1-20.  doi: 10.1016/j.physd.2006.07.024.  Google Scholar

Figure 1.  Lines with $\xi = \text{constant}$ (full), and with $\eta = \text{constant}$ (dashed) in the $(j,\tau)$ plane. The values used for these parameters are the following, in counterclockwise direction: $\xi = 5.0, 2.0, 1.0, 0.5, 0.0, -0.3, -0.5, -0.7, -0.9,$ and $\eta = 1/4,$ $1/3,$ $1/2,$ $1,$ $2,$ $3,$ $4$.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Graph of the similarity profile $\Phi_{2,n}(\xi)$ for different values of $n$.
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212
[2]

Jan Prüss, Laurent Pujo-Menjouet, G.F. Webb, Rico Zacher. Analysis of a model for the dynamics of prions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 225-235. doi: 10.3934/dcdsb.2006.6.225
[3]

Tobias Geiger, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of ODEs with state suprema. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021012
[4]

Juan Manuel Pastor, Javier García-Algarra, Javier Galeano, José María Iriondo, José J. Ramasco. A simple and bounded model of population dynamics for mutualistic networks. Networks & Heterogeneous Media, 2015, 10 (1) : 53-70. doi: 10.3934/nhm.2015.10.53
[5]

Yu Jin, Xiao-Qiang Zhao. The spatial dynamics of a Zebra mussel model in river environments. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 1991-2010. doi: 10.3934/dcdsb.2020362
[6]

Sohana Jahan. Discriminant analysis of regularized multidimensional scaling. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 255-267. doi: 10.3934/naco.2020024
[7]

Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183
[8]

Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301
[9]

Mingxin Wang, Qianying Zhang. Dynamics for the diffusive Leslie-Gower model with double free boundaries. Discrete & Continuous Dynamical Systems, 2018, 38 (5) : 2591-2607. doi: 10.3934/dcds.2018109
[10]

Wei Wang, Wanbiao Ma, Xiulan Lai. Sufficient conditions for global dynamics of a viral infection model with nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3989-4011. doi: 10.3934/dcdsb.2020271
[11]

Weiyi Zhang, Zuhan Liu, Ling Zhou. Dynamics of a nonlocal diffusive logistic model with free boundaries in time periodic environment. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3767-3784. doi: 10.3934/dcdsb.2020256
[12]

Anton Schiela, Julian Ortiz. Second order directional shape derivatives of integrals on submanifolds. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021017
[13]

Haibo Cui, Haiyan Yin. Convergence rate of solutions toward stationary solutions to the isentropic micropolar fluid model in a half line. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2899-2920. doi: 10.3934/dcdsb.2020210
[14]

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
[15]

Renhao Cui. Asymptotic profiles of the endemic equilibrium of a reaction-diffusion-advection SIS epidemic model with saturated incidence rate. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2997-3022. doi: 10.3934/dcdsb.2020217
[16]

Christoforidou Amalia, Christian-Oliver Ewald. A lattice method for option evaluation with regime-switching asset correlation structure. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1729-1752. doi: 10.3934/jimo.2020042
[17]

Wen-Bin Yang, Yan-Ling Li, Jianhua Wu, Hai-Xia Li. Dynamics of a food chain model with ratio-dependent and modified Leslie-Gower functional responses. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2269-2290. doi: 10.3934/dcdsb.2015.20.2269
[18]

Zhikun She, Xin Jiang. Threshold dynamics of a general delayed within-host viral infection model with humoral immunity and two modes of virus transmission. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3835-3861. doi: 10.3934/dcdsb.2020259
[19]

Marzia Bisi, Maria Groppi, Giorgio Martalò, Romina Travaglini. Optimal control of leachate recirculation for anaerobic processes in landfills. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2957-2976. doi: 10.3934/dcdsb.2020215
[20]

Prabir Panja, Soovoojeet Jana, Shyamal kumar Mondal. Dynamics of a stage structure prey-predator model with ratio-dependent functional response and anti-predator behavior of adult prey. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 391-405. doi: 10.3934/naco.2020033

2019 Impact Factor: 1.311

Tools

Metrics

  • PDF downloads (96)
  • HTML views (139)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences