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December  2021, 26(12): 6387-6403. doi: 10.3934/dcdsb.2021024

On the dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions

Fang Li 1, and  Bo You 2,,

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  December 2021 Early access  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.

Keywords: Global attractor, Cahn-Hilliard-Brinkman system, Fractal dimension, Dynamic boundary conditions, Dissipativity, The $ \ell $-trajectory method.
Mathematics Subject Classification: 35B41, 35Q35, 37L30.
Citation: Fang Li, Bo You. On the dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6387-6403. 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.

[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.

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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.

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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.

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V. Chepyzhov and M. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, 2002.

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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.

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A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations. Research in Applied Mathematics, Providence, RI: Masson, 1994.

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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.

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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.

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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.

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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).

[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).

[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.

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