-
Previous Article
Recurrent solutions of the Schrödinger-KdV system with boundary forces
- DCDS-B Home
- This Issue
-
Next Article
Propagation phenomena for a criss-cross infection model with non-diffusive susceptible population in periodic media
On the dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions
1. | School of Mathematics and Statistics, Xidian University, Xi'an, 710126, China |
2. | School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049, China |
The objective of this paper is to study the fractal dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions. Inspired by the idea of the $ \ell $-trajectory method, we prove the existence of a finite dimensional global attractor in an auxiliary normed space for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions and estimate the fractal dimension of the global attractor in the original phase space for this system by defining a Lipschitz mapping from the auxiliary normed space into the original phase space.
References:
[1] |
F. Abergel, Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains, Journal of Differential Equations, 83 (1990), 85-108.
doi: 10.1016/0022-0396(90)90070-6. |
[2] |
A. V. Babin and M. I. Vishik, Attractors of partial differential evolution equations and estimates of their dimension, Russian Mathematical Surveys, 38 (1983), 133-187. |
[3] |
F. Balibrea and J. Valero, Estimates of dimension of attractors of reaction-diffusion equations in the non-differentiable case, Comptes Rendus de l Academie des Sciences-I: Mathematics, 325 (1997). 759-764.
doi: 10.1016/S0764-4442(97)80056-0. |
[4] |
S. Bosia, M. Conti and M. Grasselli, On the Cahn-Hilliard-Brinkman system, Communications in Mathematical Sciences, 13 (2015), 1541-1567.
doi: 10.4310/CMS.2015.v13.n6.a9. |
[5] |
H. C. Brinkman, A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles, Flow, Turbulence and Combustion volume, 1 (1949), 27-36.
doi: 10.1007/BF02120313. |
[6] |
V. Chepyzhov and M. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, 2002. |
[7] |
V. V. Chepyzhov and A. A. Ilyin, A note on the fractal dimension of attractors of dissipative dynamical systems, Nonlinear Analysis, 44 (2001), 811-819.
doi: 10.1016/S0362-546X(99)00309-0. |
[8] |
R. Chill, E. Fasangova and J. Pruss, Convergence to steady states of solutions of the Cahn-Hilliard equation with dynamic boundary conditions, Mathematische Nachrichten, 279 (2006), 1448-1462.
doi: 10.1002/mana.200410431. |
[9] |
C. Collins, J. Shen and S. M. Wise, An efficient, energy stable scheme for the Cahn-Hilliard-Brinkman system, Communications in Computational Physics, 13 (2013), 929-957.
doi: 10.4208/cicp.171211.130412a. |
[10] |
A. E. Diegel, X. H. Feng and S. M. Wise, Analysis of a mixed finite element method for a Cahn-Hilliard-Darcy-Stokes system, SIAM Journal on Numerical Analysis, 53 (2015), 127-152.
doi: 10.1137/130950628. |
[11] |
A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations. Research in Applied Mathematics, Providence, RI: Masson, 1994. |
[12] |
M. Efendiev and A. Miranville, The dimension of the global attractor for dissipative reaction-diffusion systems, Applied Mathematics Letters, 16 (2003), 351-355.
doi: 10.1016/S0893-9659(03)80056-3. |
[13] |
M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\mathbb{R}^3$, C. R. Math. Acad. Sci. Paris, 330 (2000). 713-718.
doi: 10.1016/S0764-4442(00)00259-7. |
[14] |
Z. H. Fan and C. K. Zhong, Attractors for parabolic equations with dynamic boundary conditions, Nonlinear Analysis, 68 (2008), 1723-1732.
doi: 10.1016/j.na.2007.01.005. |
[15] |
C. G. Gal, Exponential attractors for a Cahn-Hilliard model in bounded domains with permeable walls, Electronic Journal of Differential Equations, 2006 (2006), 1-23. |
[16] |
C. G. Gal, Global well-posedness for the non-isothermal Cahn-Hilliard equation with dynamic boundary conditions, Advances in Differential Equations, 12 (2007), 1241-1274. |
[17] |
C. G. Gal, On a class of degenerate parabolic equations with dynamic boundary conditions, Journal of Differential Equations, 253 (2012), 126-166.
doi: 10.1016/j.jde.2012.02.010. |
[18] |
M. Grasselli, D. Pražák and G. Schimperna, Attractors for nonlinear reaction-diffusion systems in unbounded domains via the method of short trajectories, Journal of Differential Equations, 249 (2010), 2287-2315.
doi: 10.1016/j.jde.2010.06.001. |
[19] |
N. Ju, The global attractor for the solutions to the three dimensional viscous primitive equations, Discrete and Continuous Dynamical Systems, 17 (2007), 159-179.
doi: 10.3934/dcds.2007.17.159. |
[20] |
O. A. Ladyzhenskaya, On the determination of minimal global attractors for Navier-Stokes equations and other partial differential equations, Uspekhi Matematicheskikh Nauk, 42 (1987), 25-60. |
[21] |
J. A. Langa and J. C. Robinson, Fractal dimension of a random invariant set, Journal de Mathématiques Pures et Appliquées, 85 (2006), 269-294.
doi: 10.1016/j.matpur.2005.08.001. |
[22] |
F. Li, C. K. Zhong and B. You, Finite-dimensional global attractor of the Cahn-Hilliard-Brinkman system, Journal of Mathematical Analysis and Applications, 434 (2016), 599-616.
doi: 10.1016/j.jmaa.2015.09.026. |
[23] |
Y. Lv and W. Wang, Limiting dynamics for stochastic wave equations, Journal of Differential Equations, 244 (2008), 1-23.
doi: 10.1016/j.jde.2007.10.009. |
[24] |
J. Málek and J. Nečas, A finite-dimensional attractor for three-dimensional flow of incompressible fluids, Journal of Differential Equations, 127 (1996), 498-518.
doi: 10.1006/jdeq.1996.0080. |
[25] |
J. Málek and D. Pražák, Finite fractal dimension of the global attractor for a class of non-newtonian fluids, Applied Mathematics Letters, 13 (2000), 105-110.
doi: 10.1016/S0893-9659(99)00152-4. |
[26] |
J. Málek and D. Pražák, Large time behavior via the method of $\ell$-trajectories, Journal of Differential Equations, 181 (2002), 243-279.
doi: 10.1006/jdeq.2001.4087. |
[27] |
A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamical boundary conditions, Mathematical Models and Methods in Applied Sciences, 28 (2005), 709-735.
doi: 10.1002/mma.590. |
[28] |
W. Ngamsaad, J. Yojina and W. Triampo, Theoretical studies of phase-separation kinetics in a Brinkman porous medium, Journal of Physics A: Mathematical and Theoretical, 43 (2010), 202001(7pp). Google Scholar |
[29] |
D. Pražák, On finite fractal dimension of the global attractor for the wave equation with nonlinear damping, Journal of Dynamics and Differential Equations, 14 (2002), 763-776.
doi: 10.1023/A:1020756426088. |
[30] |
D. Pražák, On the dimension of the attractor for the wave equation with nonlinear damping, Communications on Pure and Applied Analysis, 4 (2005), 165-174.
doi: 10.3934/cpaa.2005.4.165. |
[31] |
J. Pruss, R. Racke and S. Zheng, Maximal regularity and asymptotic behavior of solutions for the Cahn-Hilliard equation with dynamic boundary conditions, Annali di Matematica Pura ed Applicata, 185 (2006), 627-648.
doi: 10.1007/s10231-005-0175-3. |
[32] |
J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic Partial Differential Equations and the Theory of Global Attractors, Cambridge University Press, 2001. |
[33] |
R. Rosa, The global attractor for the 2D Navier-Stokes flow on some unbounded domains, Nonlinear Analysis, 32 (1998), 71-85.
doi: 10.1016/S0362-546X(97)00453-7. |
[34] |
J. Simon, Compact sets in the space $l^p(0, t;b)$, Annali di Matematica Pura ed Applicata, 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
[35] |
R. Temam, Infinite-dimensional Systems in Mechanics and Physics, Springer-Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[36] |
B. You and F. Li, Well-posedness and global attractor of the Cahn-Hilliard-Brinkman system with dynamic boundary conditions, Dynamics of Partial Differential Equations, 13 (2016), 75-90.
doi: 10.4310/DPDE.2016.v13.n1.a4. |
[37] |
B. You and C. K. Zhong, Global attractors for $p$-laplacian equations with dynamic flux boundary conditions, Advanced Nonlinear Studies, 13 (2013), 391-410.
doi: 10.1515/ans-2013-0208. |
show all references
References:
[1] |
F. Abergel, Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains, Journal of Differential Equations, 83 (1990), 85-108.
doi: 10.1016/0022-0396(90)90070-6. |
[2] |
A. V. Babin and M. I. Vishik, Attractors of partial differential evolution equations and estimates of their dimension, Russian Mathematical Surveys, 38 (1983), 133-187. |
[3] |
F. Balibrea and J. Valero, Estimates of dimension of attractors of reaction-diffusion equations in the non-differentiable case, Comptes Rendus de l Academie des Sciences-I: Mathematics, 325 (1997). 759-764.
doi: 10.1016/S0764-4442(97)80056-0. |
[4] |
S. Bosia, M. Conti and M. Grasselli, On the Cahn-Hilliard-Brinkman system, Communications in Mathematical Sciences, 13 (2015), 1541-1567.
doi: 10.4310/CMS.2015.v13.n6.a9. |
[5] |
H. C. Brinkman, A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles, Flow, Turbulence and Combustion volume, 1 (1949), 27-36.
doi: 10.1007/BF02120313. |
[6] |
V. Chepyzhov and M. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, 2002. |
[7] |
V. V. Chepyzhov and A. A. Ilyin, A note on the fractal dimension of attractors of dissipative dynamical systems, Nonlinear Analysis, 44 (2001), 811-819.
doi: 10.1016/S0362-546X(99)00309-0. |
[8] |
R. Chill, E. Fasangova and J. Pruss, Convergence to steady states of solutions of the Cahn-Hilliard equation with dynamic boundary conditions, Mathematische Nachrichten, 279 (2006), 1448-1462.
doi: 10.1002/mana.200410431. |
[9] |
C. Collins, J. Shen and S. M. Wise, An efficient, energy stable scheme for the Cahn-Hilliard-Brinkman system, Communications in Computational Physics, 13 (2013), 929-957.
doi: 10.4208/cicp.171211.130412a. |
[10] |
A. E. Diegel, X. H. Feng and S. M. Wise, Analysis of a mixed finite element method for a Cahn-Hilliard-Darcy-Stokes system, SIAM Journal on Numerical Analysis, 53 (2015), 127-152.
doi: 10.1137/130950628. |
[11] |
A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations. Research in Applied Mathematics, Providence, RI: Masson, 1994. |
[12] |
M. Efendiev and A. Miranville, The dimension of the global attractor for dissipative reaction-diffusion systems, Applied Mathematics Letters, 16 (2003), 351-355.
doi: 10.1016/S0893-9659(03)80056-3. |
[13] |
M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\mathbb{R}^3$, C. R. Math. Acad. Sci. Paris, 330 (2000). 713-718.
doi: 10.1016/S0764-4442(00)00259-7. |
[14] |
Z. H. Fan and C. K. Zhong, Attractors for parabolic equations with dynamic boundary conditions, Nonlinear Analysis, 68 (2008), 1723-1732.
doi: 10.1016/j.na.2007.01.005. |
[15] |
C. G. Gal, Exponential attractors for a Cahn-Hilliard model in bounded domains with permeable walls, Electronic Journal of Differential Equations, 2006 (2006), 1-23. |
[16] |
C. G. Gal, Global well-posedness for the non-isothermal Cahn-Hilliard equation with dynamic boundary conditions, Advances in Differential Equations, 12 (2007), 1241-1274. |
[17] |
C. G. Gal, On a class of degenerate parabolic equations with dynamic boundary conditions, Journal of Differential Equations, 253 (2012), 126-166.
doi: 10.1016/j.jde.2012.02.010. |
[18] |
M. Grasselli, D. Pražák and G. Schimperna, Attractors for nonlinear reaction-diffusion systems in unbounded domains via the method of short trajectories, Journal of Differential Equations, 249 (2010), 2287-2315.
doi: 10.1016/j.jde.2010.06.001. |
[19] |
N. Ju, The global attractor for the solutions to the three dimensional viscous primitive equations, Discrete and Continuous Dynamical Systems, 17 (2007), 159-179.
doi: 10.3934/dcds.2007.17.159. |
[20] |
O. A. Ladyzhenskaya, On the determination of minimal global attractors for Navier-Stokes equations and other partial differential equations, Uspekhi Matematicheskikh Nauk, 42 (1987), 25-60. |
[21] |
J. A. Langa and J. C. Robinson, Fractal dimension of a random invariant set, Journal de Mathématiques Pures et Appliquées, 85 (2006), 269-294.
doi: 10.1016/j.matpur.2005.08.001. |
[22] |
F. Li, C. K. Zhong and B. You, Finite-dimensional global attractor of the Cahn-Hilliard-Brinkman system, Journal of Mathematical Analysis and Applications, 434 (2016), 599-616.
doi: 10.1016/j.jmaa.2015.09.026. |
[23] |
Y. Lv and W. Wang, Limiting dynamics for stochastic wave equations, Journal of Differential Equations, 244 (2008), 1-23.
doi: 10.1016/j.jde.2007.10.009. |
[24] |
J. Málek and J. Nečas, A finite-dimensional attractor for three-dimensional flow of incompressible fluids, Journal of Differential Equations, 127 (1996), 498-518.
doi: 10.1006/jdeq.1996.0080. |
[25] |
J. Málek and D. Pražák, Finite fractal dimension of the global attractor for a class of non-newtonian fluids, Applied Mathematics Letters, 13 (2000), 105-110.
doi: 10.1016/S0893-9659(99)00152-4. |
[26] |
J. Málek and D. Pražák, Large time behavior via the method of $\ell$-trajectories, Journal of Differential Equations, 181 (2002), 243-279.
doi: 10.1006/jdeq.2001.4087. |
[27] |
A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamical boundary conditions, Mathematical Models and Methods in Applied Sciences, 28 (2005), 709-735.
doi: 10.1002/mma.590. |
[28] |
W. Ngamsaad, J. Yojina and W. Triampo, Theoretical studies of phase-separation kinetics in a Brinkman porous medium, Journal of Physics A: Mathematical and Theoretical, 43 (2010), 202001(7pp). Google Scholar |
[29] |
D. Pražák, On finite fractal dimension of the global attractor for the wave equation with nonlinear damping, Journal of Dynamics and Differential Equations, 14 (2002), 763-776.
doi: 10.1023/A:1020756426088. |
[30] |
D. Pražák, On the dimension of the attractor for the wave equation with nonlinear damping, Communications on Pure and Applied Analysis, 4 (2005), 165-174.
doi: 10.3934/cpaa.2005.4.165. |
[31] |
J. Pruss, R. Racke and S. Zheng, Maximal regularity and asymptotic behavior of solutions for the Cahn-Hilliard equation with dynamic boundary conditions, Annali di Matematica Pura ed Applicata, 185 (2006), 627-648.
doi: 10.1007/s10231-005-0175-3. |
[32] |
J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic Partial Differential Equations and the Theory of Global Attractors, Cambridge University Press, 2001. |
[33] |
R. Rosa, The global attractor for the 2D Navier-Stokes flow on some unbounded domains, Nonlinear Analysis, 32 (1998), 71-85.
doi: 10.1016/S0362-546X(97)00453-7. |
[34] |
J. Simon, Compact sets in the space $l^p(0, t;b)$, Annali di Matematica Pura ed Applicata, 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
[35] |
R. Temam, Infinite-dimensional Systems in Mechanics and Physics, Springer-Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[36] |
B. You and F. Li, Well-posedness and global attractor of the Cahn-Hilliard-Brinkman system with dynamic boundary conditions, Dynamics of Partial Differential Equations, 13 (2016), 75-90.
doi: 10.4310/DPDE.2016.v13.n1.a4. |
[37] |
B. You and C. K. Zhong, Global attractors for $p$-laplacian equations with dynamic flux boundary conditions, Advanced Nonlinear Studies, 13 (2013), 391-410.
doi: 10.1515/ans-2013-0208. |
[1] |
Franck Davhys Reval Langa, Morgan Pierre. A doubly splitting scheme for the Caginalp system with singular potentials and dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 653-676. doi: 10.3934/dcdss.2020353 |
[2] |
Pierluigi Colli, Gianni Gilardi, Gabriela Marinoschi. Solvability and sliding mode control for the viscous Cahn–Hilliard system with a possibly singular potential. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020051 |
[3] |
Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (1) : 301-317. doi: 10.3934/cpaa.2020267 |
[4] |
Vandana Sharma. Global existence and uniform estimates of solutions to reaction diffusion systems with mass transport type boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021001 |
[5] |
Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020345 |
[6] |
Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021015 |
[7] |
Tomáš Roubíček. Cahn-Hilliard equation with capillarity in actual deforming configurations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 41-55. doi: 10.3934/dcdss.2020303 |
[8] |
Hussein Fakih, Ragheb Mghames, Noura Nasreddine. On the Cahn-Hilliard equation with mass source for biological applications. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020277 |
[9] |
Wenjun Liu, Hefeng Zhuang. Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 907-942. doi: 10.3934/dcdsb.2020147 |
[10] |
João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138 |
[11] |
Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213 |
[12] |
Andrea Giorgini, Roger Temam, Xuan-Truong Vu. The Navier-Stokes-Cahn-Hilliard equations for mildly compressible binary fluid mixtures. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 337-366. doi: 10.3934/dcdsb.2020141 |
[13] |
Erica Ipocoana, Andrea Zafferi. Further regularity and uniqueness results for a non-isothermal Cahn-Hilliard equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020289 |
[14] |
Chao Xing, Zhigang Pan, Quan Wang. Stabilities and dynamic transitions of the Fitzhugh-Nagumo system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 775-794. doi: 10.3934/dcdsb.2020134 |
[15] |
Marcos C. Mota, Regilene D. S. Oliveira. Dynamic aspects of Sprott BC chaotic system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1653-1673. doi: 10.3934/dcdsb.2020177 |
[16] |
Tengfei Yan, Qunying Liu, Bowen Dou, Qing Li, Bowen Li. An adaptive dynamic programming method for torque ripple minimization of PMSM. Journal of Industrial & Management Optimization, 2021, 17 (2) : 827-839. doi: 10.3934/jimo.2019136 |
[17] |
Roland Schnaubelt, Martin Spitz. Local wellposedness of quasilinear Maxwell equations with absorbing boundary conditions. Evolution Equations & Control Theory, 2021, 10 (1) : 155-198. doi: 10.3934/eect.2020061 |
[18] |
Kuntal Bhandari, Franck Boyer. Boundary null-controllability of coupled parabolic systems with Robin conditions. Evolution Equations & Control Theory, 2021, 10 (1) : 61-102. doi: 10.3934/eect.2020052 |
[19] |
Qianqian Hou, Tai-Chia Lin, Zhi-An Wang. On a singularly perturbed semi-linear problem with Robin boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 401-414. doi: 10.3934/dcdsb.2020083 |
[20] |
Wenrui Hao, King-Yeung Lam, Yuan Lou. Ecological and evolutionary dynamics in advective environments: Critical domain size and boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 367-400. doi: 10.3934/dcdsb.2020283 |
2019 Impact Factor: 1.27
Tools
Article outline
[Back to Top]