Convective nonlocal Cahn-Hilliard equations with reaction terms

Francesco Della Porta; Maurizio Grasselli

doi:10.3934/dcdsb.2015.20.1529

Article Contents

2015, Volume 20, Issue 5: 1529-1553. Doi: 10.3934/dcdsb.2015.20.1529

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Convective nonlocal Cahn-Hilliard equations with reaction terms

Francesco Della Porta^1, and
Maurizio Grasselli^2,

1.
Mathematical Institute, University of Oxford, Oxford OX2 6GG
2.
Dipartimento di Matematica, Politecnico di Milano, 20133 Milano

Received: May 2014

Revised: January 2015

Published: July 2015

Abstract / Introduction Related Papers Cited by

Abstract

We introduce and analyze the nonlocal variants of two Cahn-Hilliard type equations with reaction terms. The first one is the so-called Cahn-Hilliard-Oono equation which models, for instance, pattern formation in diblock-copolymers as well as in binary alloys with induced reaction and type-I superconductors. The second one is the Cahn-Hilliard type equation introduced by Bertozzi et al. to describe image inpainting. Here we take a free energy functional which accounts for nonlocal interactions. Our choice is motivated by the work of Giacomin and Lebowitz who showed that the rigorous physical derivation of the Cahn-Hilliard equation leads to consider nonlocal functionals. The equations also have a transport term with a given velocity field and are subject to a homogenous Neumann boundary condition for the chemical potential, i.e., the first variation of the free energy functional. We first establish the well-posedness of the corresponding initial and boundary value problems in a weak setting. Then we consider such problems as dynamical systems and we show that they have bounded absorbing sets and global attractors.

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Mathematics Subject Classification: Primary: 37L30, 45K05, 76T99, 80A32.

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References

[1]	A. C. Aristotelous, O. Karakashian and S. M. Wise, A mixed discontinuous Galerkin, convex splitting scheme for a modified Cahn-Hilliard equation and an efficient nonlinear multigrid solver, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2211-2238.doi: 10.3934/dcdsb.2013.18.2211.
[2]	M. Bahiana and Y. Oono, Cell dynamical system approach to block copolymers, Phys. Rev. A, 41 (1990), 6763-6771.doi: 10.1103/PhysRevA.41.6763.
[3]	J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equation, J. Nonlinear Sci., 7 (1997), 475-502 (Erratum, J. Nonlinear Sci. 8 (1998), p233).doi: 10.1007/s003329900037.
[4]	P. W. Bates and J. Han, The Neumann boundary problem for a nonlocal Cahn-Hilliard equation, J. Differential Equations, 212 (2005), 235-277.doi: 10.1016/j.jde.2004.07.003.
[5]	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.
[6]	A. L. Bertozzi, S. Esedo$\overlineg$lu 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.
[7]	A. L. Bertozzi, S. Esedo$\overlineg$lu and A. Gillette, Analysis of a two-scale Cahn-Hilliard model for binary image inpainting, Multiscale Model. Simul., 6 (2007), 913-936.doi: 10.1137/060660631.
[8]	S. Bosia, M. Grasselli and A. Miranville, On the longtime behavior of a 2D hydrodynamic model for chemically reacting binary fluid mixtures, Math. Methods Appl. Sci., 37 (2014), 726-743.doi: 10.1002/mma.2832.
[9]	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.
[10]	J. W. Cahn, On spinodal decomposition, Acta Met., 9 (1961), 795-801.doi: 10.1002/9781118788295.ch11.
[11]	R. Choksi, M. A. Peletier and J. F. Williams, On the phase diagram for microphase separation of diblock copolymers: An approach via a nonlocal Cahn-Hilliard functional, SIAM J. Appl. Math., 69 (2009), 1712-1738.doi: 10.1137/080728809.
[12]	R. Choksi, Scaling laws in microphase separation of diblock copolymers, J. Nonlinear Sci., 11 (2001), 223-236.doi: 10.1007/s00332-001-0456-y.
[13]	R. Choksi and X. Ren, On the derivation of a density functional theory for microphase separation of diblock copolymers, J. Stat. Phys., 113 (2003), 151-176.doi: 10.1023/A:1025722804873.
[14]	R. Choksi, M. Maras and J. F. Williams, 2D phase diagram for minimizers of a Cahn-Hilliard functional with long-range interactions, SIAM J. Appl. Dyn. Syst., 10 (2011), 1344-1362.doi: 10.1137/100784497.
[15]	L. Cherfils, H. Fakih and A. Miranville, Finite-dimensional attractors for the Bertozzi-Esedo$\overlineg$lu-Gillette-Cahn-Hilliard equation in image inpainting, Inverse Probl. Imaging, 9 (2015), 105-125.
[16]	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.
[17]	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.
[18]	P. Colli, S. Frigeri and M. Grasselli, Global existence of weak solutions to a nonlocal Cahn-Hilliard-Navier-Stokes system, J. Math. Anal. Appl., 386 (2012), 428-444.doi: 10.1016/j.jmaa.2011.08.008.
[19]	A. Debussche and L. Dettori, On the Cahn-Hilliard equation with a logarithmic free energy, Nonlinear Anal., 24 (1995), 1491-1514.doi: 10.1016/0362-546X(94)00205-V.
[20]	C. M. Elliott and S. Zheng, On the Cahn-Hilliard equation, Arch. Ration. Mech. Anal., 96 (1986), 339-357.doi: 10.1007/BF00251803.
[21]	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.
[22]	P. C. Fife, Models for phase separation and their mathematics, Electron. J. Differential Equations, 48 (2000), 26 pp.
[23]	S. Frigeri, C. G. Gal and M. Grasselli, On nonlocal Cahn-Hilliard-Navier-Stokes systems in two dimensions, preprint, WIAS Preprint, 1923 (2014), 34 pp.
[24]	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.
[25]	S. Frigeri and M. Grasselli, Nonlocal Cahn-Hilliard-Navier-Stokes systems with singular potentials, Dyn. Partial Differ. Equ., 9 (2012), 273-304.doi: 10.4310/DPDE.2012.v9.n4.a1.
[26]	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.
[27]	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.
[28]	G. Giacomin and J. L. Lebowitz, Exact macroscopic description of phase segregation in model alloys with long range interactions, Phys. Rev. Lett., 76 (1996), 1094-1097.doi: 10.1103/PhysRevLett.76.1094.
[29]	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.
[30]	G. Giacomin and J. L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions. II. Interface motion, SIAM J. Appl. Math., 58 (1998), 1707-1729.doi: 10.1137/S0036139996313046.
[31]	S. C. Glotzer, E. A. Di Marzio and M. Muthukumar, Reaction-controlled morphology of phase separating mixtures, Phys. Rev. Lett., 74 (1995), 2034-2037.doi: 10.1103/PhysRevLett.74.2034.
[32]	Y. Huo, H. Zhang and Y. Yang, Effects of reversible chemical reaction on morphology and domain growth of phase separating binary mixtures with viscosity difference, Macromol. Theory Simul., 13 (2004), 280-289.doi: 10.1002/mats.200300021.
[33]	Y. Huo, X. Jiang, H. Zhang and Y. Yang, Hydrodynamic effects on phase separation of binary mixtures with reversible chemical reaction, J. Chem. Phys., 118 (2003), 9830-9837.doi: 10.1063/1.1571511.
[34]	T. P. Lodge, Block copolymers: past successes and future challenges, Macromol. Chem. Phys., 204 (2003), 265-273.doi: 10.1002/macp.200290073.
[35]	S.-O. 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.
[36]	S.-O. Londen and H. Petzeltová, Convergence of solutions of a non-local phase-field system, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 653-670.
[37]	P. Mansky, P. Chaikin and E. L. Thomas, Monolayer films of diblock copolymer microdomains for nanolithographic applications, J. Mater. Sci., 30 (1995), 1987-1992.doi: 10.1007/BF00353023.
[38]	S. Melchionna and E. Rocca, On a nonlocal Cahn-Hilliard equation with a reaction term, Adv. Math. Sci. Appl., 24 (2014), 461-497.
[39]	A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, Handbook of differential equations: evolutionary equations, IV (2008), 103-200.doi: 10.1016/S1874-5717(08)00003-0.
[40]	A. Miranville, Asymptotic behavior of the Cahn-Hilliard-Oono equation, J. Appl. Anal. Comput., 1 (2011), 523-536.
[41]	A. Miranville, Asymptotic behaviour of a generalized Cahn-Hilliard equation with a proliferation term, Appl. Anal., 92 (2013), 1308-1321.doi: 10.1080/00036811.2012.671301.
[42]	C. B. Muratov, Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions, Comm. Math. Phys., 299 (2010), 45-87.doi: 10.1007/s00220-010-1094-8.
[43]	C. B. Muratov, Theory of domain patterns in systems with long-range interactions of Coulomb type, Phys. Rev. E, 66 (2002), 066108, 25 pp.doi: 10.1103/PhysRevE.66.066108.
[44]	B. Nicolaenko, B. Scheurer and R. Temam, Some global dynamical properties of a class of pattern formation equations, Comm. Partial Differential Equations, 14 (1989), 245-297.doi: 10.1080/03605308908820597.
[45]	Y. Nishiura and I. Ohnishi, Some mathematical aspects of the microphase separation in diblock copolymers, Phys. D, 84 (1995), 31-39.doi: 10.1016/0167-2789(95)00005-O.
[46]	A. Novick-Cohen, The Cahn-Hilliard equation: Mathematical and modeling perspectives, Adv. Math. Sci. Appl., 8 (1998), 965-985.
[47]	A. Novick-Cohen, The Cahn-Hilliard equation, Handbook of differential equations: Evolutionary equations, IV (2008), 201-228.doi: 10.1016/S1874-5717(08)00004-2.
[48]	T. Ohta and K. Kawasaki, Equilibrium morphology of block copolymer melts, Macromolecules, 19 (1986), 2621-2632.doi: 10.1021/ma00164a028.
[49]	Y. Oono and S. Puri, Computationally Efficient Modeling of Ordering of Quenched Phases, Phys. Rev. Lett., 58 (1987), 836-839.doi: 10.1103/PhysRevLett.58.836.
[50]	G. Schimperna, Global attractors for Cahn-Hilliard equations with nonconstant mobility, Nonlinearity, 20 (2007), 2365-2387.doi: 10.1088/0951-7715/20/10/006.
[51]	S. Villain Guillot, 1D Cahn-Hilliard equation for modulated phase systems, J. Phys. A, 43 (2010).
[52]	S. Walheim, E. Schaeffer, J. Mlynek and U. Steiner, Nanophase-separated polymer films as high-performance antireflection coatings, Science, 283 (1999), 520-522.doi: 10.1126/science.283.5401.520.
[53]	S. Zheng, Asymptotic behavior of solution to the Cahn-Hilliard equation, Appl. Anal., 23 (1986), 165-184.doi: 10.1080/00036818608839639.