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August
2018, 38(8): 3765-3788.
doi: 10.3934/dcds.2018163
Long-time behavior of a nonlocal Cahn-Hilliard equation with reaction
| 1. | Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8-10, 1040, Vienna, Austria |
| 2. | Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria |
In this paper we study the long-time behavior of a nonlocal Cahn-Hilliard system with singular potential, degenerate mobility, and a reaction term. In particular, we prove the existence of a global attractor with finite fractal dimension, the existence of an exponential attractor, and convergence to equilibria for two physically relevant classes of reaction terms.
Keywords:
Nonlocal Cahn-Hilliard equation,
reaction term,
global attractor,
exponential attractor,
convergence to equilibria.
Mathematics Subject Classification:
Primary: 37L30; Secondary: 45K05.
Citation:
Annalisa Iuorio, Stefano Melchionna. Long-time behavior of a nonlocal Cahn-Hilliard equation with reaction. Discrete and Continuous Dynamical Systems,
2018, 38
(8)
: 3765-3788.
doi: 10.3934/dcds.2018163
![]()
References:
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H. Abels and M. Wilke,
Convergence to equilibrium for the Cahn-Hilliard equation with a logarithmic free energy, Nonlinear Anal., 67 (2007), 3176-3193.
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G. Alberti and G. Bellettini,
A non-local anisotropic model for phase transitions: Asymptotic behaviour of rescaled energies, European J. Appl. Math., 9 (1998), 261-284.
doi: 10.1017/S0956792598003453. |
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M. Bahiana and Y. Oono, Cell dynamical system approach to block copolymers, Phys. Rev. A, 41 (1990), p6763. |
| [4] |
P. W. Bates and J. Han,
The Dirichlet boundary problem for a nonlocal Cahn-Hilliard equation, J. Math. Anal. Appl., 311 (2005), 289-312.
doi: 10.1016/j.jmaa.2005.02.041. |
| [5] |
A. L. Bertozzi, S. Esedoḡlu and A. Gillette,
Inpainting of binary images using the Cahn-Hilliard equation, IEEE Trans. Image Process., 16 (2007), 285-291.
doi: 10.1109/TIP.2006.887728. |
| [6] |
J. W. Cahn,
On spinodal decomposition, Acta Metall., 9 (1961), 795-801.
doi: 10.1002/9781118788295.ch11. |
| [7] |
J. W. Cahn and J. E. Hilliard,
Free energy of a nonuniform system. Ⅰ. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.
doi: 10.1002/9781118788295.ch4. |
| [8] |
L. Cherfils, H. Fakih and A. Miranville,
Finite-dimensional attractors for the Bertozzi-Esedoglu-Gillette-Cahn-Hilliard equation in image inpainting, Inverse Probl. Imaging, 9 (2015), 105-125.
doi: 10.3934/ipi.2015.9.105. |
| [9] |
L. Cherfils, H. Fakih and A. Miranville,
On the Bertozzi-Esedoglu-Gillette-Cahn-Hilliard equation with logarithmic nonlinear terms, SIAM J. Imaging Sci., 8 (2015), 1123-1140.
doi: 10.1137/140985627. |
| [10] |
L. Cherfils, A. Miranville and S. Zelik,
The Cahn-Hilliard equation with logarithmic potentials, Milan J. Math., 79 (2011), 561-596.
doi: 10.1007/s00032-011-0165-4. |
| [11] |
L. Cherfils, A. Miranville and S. Zelik,
On a generalized Cahn-Hilliard equation with biological applications, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2013-2026.
doi: 10.3934/dcdsb.2014.19.2013. |
| [12] |
F. Della Porta and M. Grasselli, Convective nonlocal Cahn-Hilliard equations with reaction terms, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1529–1553, arXiv: math/0605406.
doi: 10.3934/dcdsb.2015.20.1529. |
| [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., 330 (2000), 713-718.
doi: 10.1016/S0764-4442(00)00259-7. |
| [14] |
C. M. Elliott and H. Garcke,
On the Cahn-Hilliard equation with degenerate mobility, SIAM J. Math. Anal., 27 (1996), 404-423.
doi: 10.1137/S0036141094267662. |
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L. C. Evans,
Partial Differential Equations, American Mathematical Society, New York, 1998. |
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H. Fakih,
A Cahn-Hilliard equation with a proliferation term for biological and chemical applications, Asymptot. Anal., 94 (2015), 71-104.
doi: 10.3233/ASY-151306. |
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E. Feireisl, F. Issard-Roch and H. Petzeltová,
A non-smooth version of the Lojasiewicz-Simon theorem with applications to non-local phase-field systems, J. Differential Equations, 199 (2004), 1-21.
doi: 10.1016/j.jde.2003.10.026. |
| [18] |
S. Frigeri and M. Grasselli,
Global and trajectory attractors for a nonlocal Cahn-Hilliard-Navier-Stokes system, J. Dynam. Differential Equations, 24 (2012), 827-856.
doi: 10.1007/s10884-012-9272-3. |
| [19] |
S. Frigeri, M. Grasselli and E. Rocca,
A diffuse interface model for two-phase incompressible flows with non-local interactions and non-constant mobility, Nonlinearity, 28 (2015), 1257-1293.
doi: 10.1088/0951-7715/28/5/1257. |
| [20] |
H. Gajewski and K. Zacharias,
On a nonlocal phase separation model, J. Math. Anal. Appl., 286 (2003), 11-31.
doi: 10.1016/S0022-247X(02)00425-0. |
| [21] |
C. G. Gal and M. Grasselli,
Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 401-436.
doi: 10.1016/j.anihpc.2009.11.013. |
| [22] |
C. G. Gal and M. Grasselli,
Longtime behavior of nonlocal Cahn-Hilliard equations, Discrete Contin. Dyn. Syst., 34 (2014), 145-179.
doi: 10.3934/dcds.2014.34.145. |
| [23] |
H. Garcke, B. Nestler and B. Stoth,
On anisotropic order parameter models for multi-phase systems and their sharp interface limits, Phys. D, 115 (1998), 87-108.
doi: 10.1016/S0167-2789(97)00227-3. |
| [24] |
H. Garcke, M. Rumpf and U. Weikard,
The Cahn-Hilliard equation with elasticity-finite element approximation and qualitative studies, Interfaces Free Bound., 3 (2001), 101-118.
doi: 10.4171/IFB/34. |
| [25] |
G. Giacomin and J. L. Lebowitz,
Phase segregation dynamics in particle systems with long range interactions. Ⅰ. Macroscopic limits, J. Stat. Phys, 87 (1997), 37-61.
doi: 10.1007/BF02181479. |
| [26] |
Z. Guan, J. S. Lowengrub, C. Wang and S. M. Wise,
Second order convex splitting schemes for periodic nonlocal Cahn-Hilliard and Allen-Cahn equations, J. Comput. Phys., 277 (2014), 48-71.
doi: 10.1016/j.jcp.2014.08.001. |
| [27] |
E. Khain and L. M. Sander, Generalized Cahn-Hilliard equation for biological applications, Phys. Rev. E, 77 (2008), 051129.
doi: 10.1103/PhysRevE.77.051129. |
| [28] |
N. Q. Le,
A Gamma-convergence approach to the Cahn-Hilliard equation, Calc. Var. Partial Differential Equations, 32 (2008), 499-522.
doi: 10.1007/s00526-007-0150-5. |
| [29] |
M. Liero and S. Reichelt, Homogenization of Cahn-Hilliard-type equations via evolutionary Γ-convergence, NoDEA Nonlinear Differential Equations Appl., 25 (2018), Art. 6, 31 pp.
doi: 10.1007/s00030-018-0495-9. |
| [30] |
S. Londen and H. Petzeltová,
Convergence of solutions of a non-local phase-field system, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 653-670.
doi: 10.3934/dcdss.2011.4.653. |
| [31] |
S. Londen and H. Petzeltová,
Regularity and separation from potential barriers for a non-local phase-field system, J. Math. Anal. Appl., 379 (2011), 724-735.
doi: 10.1016/j.jmaa.2011.02.003. |
| [32] |
S. Melchionna and E. Rocca,
On a nonlocal Cahn-Hilliard equation with a reaction term, Adv. Math. Sci. Appl., 24 (2014), 461-497.
|
| [33] |
A. Miranville,
Asymptotic behavior of the Cahn-Hilliard-Oono equation, J. Appl. Anal. Comput., 1 (2011), 523-536.
|
| [34] |
L. Modica and S. Mortola,
Un esempio di Γ-convergenza, Boll. Unione Mat. Ital. B (5), 14 (1977), 285-299.
|
| [35] |
L. Nirenberg, On elliptic partial differential equations, in Il Principio Di Minimo E Sue Applicazioni Alle Equazioni Funzionali), Springer, (2011), 1–48.
doi: 10.1007/978-3-642-10926-3_1. |
| [36] |
J. C. Robinson, Infinite-dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Vol. 28, Cambridge University Press, 2001.
doi: 10.1007/978-94-010-0732-0. |
show all references
References:
| [1] |
H. Abels and M. Wilke,
Convergence to equilibrium for the Cahn-Hilliard equation with a logarithmic free energy, Nonlinear Anal., 67 (2007), 3176-3193.
doi: 10.1016/j.na.2006.10.002. |
| [2] |
G. Alberti and G. Bellettini,
A non-local anisotropic model for phase transitions: Asymptotic behaviour of rescaled energies, European J. Appl. Math., 9 (1998), 261-284.
doi: 10.1017/S0956792598003453. |
| [3] |
M. Bahiana and Y. Oono, Cell dynamical system approach to block copolymers, Phys. Rev. A, 41 (1990), p6763. |
| [4] |
P. W. Bates and J. Han,
The Dirichlet boundary problem for a nonlocal Cahn-Hilliard equation, J. Math. Anal. Appl., 311 (2005), 289-312.
doi: 10.1016/j.jmaa.2005.02.041. |
| [5] |
A. L. Bertozzi, S. Esedoḡlu and A. Gillette,
Inpainting of binary images using the Cahn-Hilliard equation, IEEE Trans. Image Process., 16 (2007), 285-291.
doi: 10.1109/TIP.2006.887728. |
| [6] |
J. W. Cahn,
On spinodal decomposition, Acta Metall., 9 (1961), 795-801.
doi: 10.1002/9781118788295.ch11. |
| [7] |
J. W. Cahn and J. E. Hilliard,
Free energy of a nonuniform system. Ⅰ. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.
doi: 10.1002/9781118788295.ch4. |
| [8] |
L. Cherfils, H. Fakih and A. Miranville,
Finite-dimensional attractors for the Bertozzi-Esedoglu-Gillette-Cahn-Hilliard equation in image inpainting, Inverse Probl. Imaging, 9 (2015), 105-125.
doi: 10.3934/ipi.2015.9.105. |
| [9] |
L. Cherfils, H. Fakih and A. Miranville,
On the Bertozzi-Esedoglu-Gillette-Cahn-Hilliard equation with logarithmic nonlinear terms, SIAM J. Imaging Sci., 8 (2015), 1123-1140.
doi: 10.1137/140985627. |
| [10] |
L. Cherfils, A. Miranville and S. Zelik,
The Cahn-Hilliard equation with logarithmic potentials, Milan J. Math., 79 (2011), 561-596.
doi: 10.1007/s00032-011-0165-4. |
| [11] |
L. Cherfils, A. Miranville and S. Zelik,
On a generalized Cahn-Hilliard equation with biological applications, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2013-2026.
doi: 10.3934/dcdsb.2014.19.2013. |
| [12] |
F. Della Porta and M. Grasselli, Convective nonlocal Cahn-Hilliard equations with reaction terms, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1529–1553, arXiv: math/0605406.
doi: 10.3934/dcdsb.2015.20.1529. |
| [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., 330 (2000), 713-718.
doi: 10.1016/S0764-4442(00)00259-7. |
| [14] |
C. M. Elliott and H. Garcke,
On the Cahn-Hilliard equation with degenerate mobility, SIAM J. Math. Anal., 27 (1996), 404-423.
doi: 10.1137/S0036141094267662. |
| [15] |
L. C. Evans,
Partial Differential Equations, American Mathematical Society, New York, 1998. |
| [16] |
H. Fakih,
A Cahn-Hilliard equation with a proliferation term for biological and chemical applications, Asymptot. Anal., 94 (2015), 71-104.
doi: 10.3233/ASY-151306. |
| [17] |
E. Feireisl, F. Issard-Roch and H. Petzeltová,
A non-smooth version of the Lojasiewicz-Simon theorem with applications to non-local phase-field systems, J. Differential Equations, 199 (2004), 1-21.
doi: 10.1016/j.jde.2003.10.026. |
| [18] |
S. Frigeri and M. Grasselli,
Global and trajectory attractors for a nonlocal Cahn-Hilliard-Navier-Stokes system, J. Dynam. Differential Equations, 24 (2012), 827-856.
doi: 10.1007/s10884-012-9272-3. |
| [19] |
S. Frigeri, M. Grasselli and E. Rocca,
A diffuse interface model for two-phase incompressible flows with non-local interactions and non-constant mobility, Nonlinearity, 28 (2015), 1257-1293.
doi: 10.1088/0951-7715/28/5/1257. |
| [20] |
H. Gajewski and K. Zacharias,
On a nonlocal phase separation model, J. Math. Anal. Appl., 286 (2003), 11-31.
doi: 10.1016/S0022-247X(02)00425-0. |
| [21] |
C. G. Gal and M. Grasselli,
Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 401-436.
doi: 10.1016/j.anihpc.2009.11.013. |
| [22] |
C. G. Gal and M. Grasselli,
Longtime behavior of nonlocal Cahn-Hilliard equations, Discrete Contin. Dyn. Syst., 34 (2014), 145-179.
doi: 10.3934/dcds.2014.34.145. |
| [23] |
H. Garcke, B. Nestler and B. Stoth,
On anisotropic order parameter models for multi-phase systems and their sharp interface limits, Phys. D, 115 (1998), 87-108.
doi: 10.1016/S0167-2789(97)00227-3. |
| [24] |
H. Garcke, M. Rumpf and U. Weikard,
The Cahn-Hilliard equation with elasticity-finite element approximation and qualitative studies, Interfaces Free Bound., 3 (2001), 101-118.
doi: 10.4171/IFB/34. |
| [25] |
G. Giacomin and J. L. Lebowitz,
Phase segregation dynamics in particle systems with long range interactions. Ⅰ. Macroscopic limits, J. Stat. Phys, 87 (1997), 37-61.
doi: 10.1007/BF02181479. |
| [26] |
Z. Guan, J. S. Lowengrub, C. Wang and S. M. Wise,
Second order convex splitting schemes for periodic nonlocal Cahn-Hilliard and Allen-Cahn equations, J. Comput. Phys., 277 (2014), 48-71.
doi: 10.1016/j.jcp.2014.08.001. |
| [27] |
E. Khain and L. M. Sander, Generalized Cahn-Hilliard equation for biological applications, Phys. Rev. E, 77 (2008), 051129.
doi: 10.1103/PhysRevE.77.051129. |
| [28] |
N. Q. Le,
A Gamma-convergence approach to the Cahn-Hilliard equation, Calc. Var. Partial Differential Equations, 32 (2008), 499-522.
doi: 10.1007/s00526-007-0150-5. |
| [29] |
M. Liero and S. Reichelt, Homogenization of Cahn-Hilliard-type equations via evolutionary Γ-convergence, NoDEA Nonlinear Differential Equations Appl., 25 (2018), Art. 6, 31 pp.
doi: 10.1007/s00030-018-0495-9. |
| [30] |
S. Londen and H. Petzeltová,
Convergence of solutions of a non-local phase-field system, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 653-670.
doi: 10.3934/dcdss.2011.4.653. |
| [31] |
S. Londen and H. Petzeltová,
Regularity and separation from potential barriers for a non-local phase-field system, J. Math. Anal. Appl., 379 (2011), 724-735.
doi: 10.1016/j.jmaa.2011.02.003. |
| [32] |
S. Melchionna and E. Rocca,
On a nonlocal Cahn-Hilliard equation with a reaction term, Adv. Math. Sci. Appl., 24 (2014), 461-497.
|
| [33] |
A. Miranville,
Asymptotic behavior of the Cahn-Hilliard-Oono equation, J. Appl. Anal. Comput., 1 (2011), 523-536.
|
| [34] |
L. Modica and S. Mortola,
Un esempio di Γ-convergenza, Boll. Unione Mat. Ital. B (5), 14 (1977), 285-299.
|
| [35] |
L. Nirenberg, On elliptic partial differential equations, in Il Principio Di Minimo E Sue Applicazioni Alle Equazioni Funzionali), Springer, (2011), 1–48.
doi: 10.1007/978-3-642-10926-3_1. |
| [36] |
J. C. Robinson, Infinite-dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Vol. 28, Cambridge University Press, 2001.
doi: 10.1007/978-94-010-0732-0. |
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