# American Institute of Mathematical Sciences

## 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

* Corresponding author: youb2013@xjtu.edu.cn(B. You)

Received  July 2020 Revised  December 2020 Published  January 2021

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.

Citation: Fang Li, Bo You. On the dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021024
##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] V. Chepyzhov and M. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, 2002.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations. Research in Applied Mathematics, Providence, RI: Masson, 1994.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [35] R. Temam, Infinite-dimensional Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] V. Chepyzhov and M. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, 2002.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations. Research in Applied Mathematics, Providence, RI: Masson, 1994.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [35] R. Temam, Infinite-dimensional Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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