American Institute of Mathematical Sciences

July  2015, 20(5): 1529-1553. doi: 10.3934/dcdsb.2015.20.1529

Convective nonlocal Cahn-Hilliard equations with reaction terms

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

Received  May 2014 Revised  January 2015 Published  May 2015

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.
Citation: Francesco Della Porta, Maurizio Grasselli. Convective nonlocal Cahn-Hilliard equations with reaction terms. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1529-1553. doi: 10.3934/dcdsb.2015.20.1529
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. doi: 10.3934/dcdsb.2013.18.2211. Google Scholar [2] M. Bahiana and Y. Oono, Cell dynamical system approach to block copolymers,, Phys. Rev. A, 41 (1990), 6763. doi: 10.1103/PhysRevA.41.6763. Google Scholar [3] J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equation,, J. Nonlinear Sci., 7 (1997), 475. doi: 10.1007/s003329900037. Google Scholar [4] P. W. Bates and J. Han, The Neumann boundary problem for a nonlocal Cahn-Hilliard equation,, J. Differential Equations, 212 (2005), 235. doi: 10.1016/j.jde.2004.07.003. Google Scholar [5] P. W. Bates and J. Han, The Dirichlet boundary problem for a nonlocal Cahn-Hilliard equation,, J. Math. Anal. Appl., 311 (2005), 289. doi: 10.1016/j.jmaa.2005.02.041. Google Scholar [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. doi: 10.1109/TIP.2006.887728. Google Scholar [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. doi: 10.1137/060660631. Google Scholar [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. doi: 10.1002/mma.2832. Google Scholar [9] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy,, J. Chem. Phys., 28 (1958), 258. doi: 10.1002/9781118788295.ch4. Google Scholar [10] J. W. Cahn, On spinodal decomposition,, Acta Met., 9 (1961), 795. doi: 10.1002/9781118788295.ch11. Google Scholar [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. doi: 10.1137/080728809. Google Scholar [12] R. Choksi, Scaling laws in microphase separation of diblock copolymers,, J. Nonlinear Sci., 11 (2001), 223. doi: 10.1007/s00332-001-0456-y. Google Scholar [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. doi: 10.1023/A:1025722804873. Google Scholar [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. doi: 10.1137/100784497. Google Scholar [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. Google Scholar [16] L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials,, Milan J. Math., 79 (2011), 561. doi: 10.1007/s00032-011-0165-4. Google Scholar [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. doi: 10.3934/dcdsb.2014.19.2013. Google Scholar [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. doi: 10.1016/j.jmaa.2011.08.008. Google Scholar [19] A. Debussche and L. Dettori, On the Cahn-Hilliard equation with a logarithmic free energy,, Nonlinear Anal., 24 (1995), 1491. doi: 10.1016/0362-546X(94)00205-V. Google Scholar [20] C. M. Elliott and S. Zheng, On the Cahn-Hilliard equation,, Arch. Ration. Mech. Anal., 96 (1986), 339. doi: 10.1007/BF00251803. Google Scholar [21] C. M. Elliott and H. Garcke, On the Cahn-Hilliard equation with degenerate mobility,, SIAM J. Math. Anal., 27 (1996), 404. doi: 10.1137/S0036141094267662. Google Scholar [22] P. C. Fife, Models for phase separation and their mathematics,, Electron. J. Differential Equations, 48 (2000). Google Scholar [23] S. Frigeri, C. G. Gal and M. Grasselli, On nonlocal Cahn-Hilliard-Navier-Stokes systems in two dimensions, preprint,, WIAS Preprint, 1923 (2014). Google Scholar [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. doi: 10.1007/s10884-012-9272-3. Google Scholar [25] S. Frigeri and M. Grasselli, Nonlocal Cahn-Hilliard-Navier-Stokes systems with singular potentials,, Dyn. Partial Differ. Equ., 9 (2012), 273. doi: 10.4310/DPDE.2012.v9.n4.a1. Google Scholar [26] H. Gajewski and K. Zacharias, On a nonlocal phase separation model,, J. Math. Anal. Appl., 286 (2003), 11. doi: 10.1016/S0022-247X(02)00425-0. Google Scholar [27] C. G. Gal and M. Grasselli, Longtime behavior of nonlocal Cahn-Hilliard equations,, Discrete Contin. Dyn. Syst., 34 (2014), 145. doi: 10.3934/dcds.2014.34.145. Google Scholar [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. doi: 10.1103/PhysRevLett.76.1094. Google Scholar [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. doi: 10.1007/BF02181479. Google Scholar [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. doi: 10.1137/S0036139996313046. Google Scholar [31] S. C. Glotzer, E. A. Di Marzio and M. Muthukumar, Reaction-controlled morphology of phase separating mixtures,, Phys. Rev. Lett., 74 (1995), 2034. doi: 10.1103/PhysRevLett.74.2034. Google Scholar [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. doi: 10.1002/mats.200300021. Google Scholar [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. doi: 10.1063/1.1571511. Google Scholar [34] T. P. Lodge, Block copolymers: past successes and future challenges,, Macromol. Chem. Phys., 204 (2003), 265. doi: 10.1002/macp.200290073. Google Scholar [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. doi: 10.1016/j.jmaa.2011.02.003. Google Scholar [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. Google Scholar [37] P. Mansky, P. Chaikin and E. L. Thomas, Monolayer films of diblock copolymer microdomains for nanolithographic applications,, J. Mater. Sci., 30 (1995), 1987. doi: 10.1007/BF00353023. Google Scholar [38] S. Melchionna and E. Rocca, On a nonlocal Cahn-Hilliard equation with a reaction term,, Adv. Math. Sci. Appl., 24 (2014), 461. Google Scholar [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. doi: 10.1016/S1874-5717(08)00003-0. Google Scholar [40] A. Miranville, Asymptotic behavior of the Cahn-Hilliard-Oono equation,, J. Appl. Anal. Comput., 1 (2011), 523. Google Scholar [41] A. Miranville, Asymptotic behaviour of a generalized Cahn-Hilliard equation with a proliferation term,, Appl. Anal., 92 (2013), 1308. doi: 10.1080/00036811.2012.671301. Google Scholar [42] C. B. Muratov, Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions,, Comm. Math. Phys., 299 (2010), 45. doi: 10.1007/s00220-010-1094-8. Google Scholar [43] C. B. Muratov, Theory of domain patterns in systems with long-range interactions of Coulomb type,, Phys. Rev. E, 66 (2002). doi: 10.1103/PhysRevE.66.066108. Google Scholar [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. doi: 10.1080/03605308908820597. Google Scholar [45] Y. Nishiura and I. Ohnishi, Some mathematical aspects of the microphase separation in diblock copolymers,, Phys. D, 84 (1995), 31. doi: 10.1016/0167-2789(95)00005-O. Google Scholar [46] A. Novick-Cohen, The Cahn-Hilliard equation: Mathematical and modeling perspectives,, Adv. Math. Sci. Appl., 8 (1998), 965. Google Scholar [47] A. Novick-Cohen, The Cahn-Hilliard equation,, Handbook of differential equations: Evolutionary equations, IV (2008), 201. doi: 10.1016/S1874-5717(08)00004-2. Google Scholar [48] T. Ohta and K. Kawasaki, Equilibrium morphology of block copolymer melts,, Macromolecules, 19 (1986), 2621. doi: 10.1021/ma00164a028. Google Scholar [49] Y. Oono and S. Puri, Computationally Efficient Modeling of Ordering of Quenched Phases,, Phys. Rev. Lett., 58 (1987), 836. doi: 10.1103/PhysRevLett.58.836. Google Scholar [50] G. Schimperna, Global attractors for Cahn-Hilliard equations with nonconstant mobility,, Nonlinearity, 20 (2007), 2365. doi: 10.1088/0951-7715/20/10/006. Google Scholar [51] S. Villain Guillot, 1D Cahn-Hilliard equation for modulated phase systems,, J. Phys. A, 43 (2010). Google Scholar [52] S. Walheim, E. Schaeffer, J. Mlynek and U. Steiner, Nanophase-separated polymer films as high-performance antireflection coatings,, Science, 283 (1999), 520. doi: 10.1126/science.283.5401.520. Google Scholar [53] S. Zheng, Asymptotic behavior of solution to the Cahn-Hilliard equation,, Appl. Anal., 23 (1986), 165. doi: 10.1080/00036818608839639. Google Scholar

show all references

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. doi: 10.3934/dcdsb.2013.18.2211. Google Scholar [2] M. Bahiana and Y. Oono, Cell dynamical system approach to block copolymers,, Phys. Rev. A, 41 (1990), 6763. doi: 10.1103/PhysRevA.41.6763. Google Scholar [3] J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equation,, J. Nonlinear Sci., 7 (1997), 475. doi: 10.1007/s003329900037. Google Scholar [4] P. W. Bates and J. Han, The Neumann boundary problem for a nonlocal Cahn-Hilliard equation,, J. Differential Equations, 212 (2005), 235. doi: 10.1016/j.jde.2004.07.003. Google Scholar [5] P. W. Bates and J. Han, The Dirichlet boundary problem for a nonlocal Cahn-Hilliard equation,, J. Math. Anal. Appl., 311 (2005), 289. doi: 10.1016/j.jmaa.2005.02.041. Google Scholar [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. doi: 10.1109/TIP.2006.887728. Google Scholar [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. doi: 10.1137/060660631. Google Scholar [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. doi: 10.1002/mma.2832. Google Scholar [9] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy,, J. Chem. Phys., 28 (1958), 258. doi: 10.1002/9781118788295.ch4. Google Scholar [10] J. W. Cahn, On spinodal decomposition,, Acta Met., 9 (1961), 795. doi: 10.1002/9781118788295.ch11. Google Scholar [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. doi: 10.1137/080728809. Google Scholar [12] R. Choksi, Scaling laws in microphase separation of diblock copolymers,, J. Nonlinear Sci., 11 (2001), 223. doi: 10.1007/s00332-001-0456-y. Google Scholar [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. doi: 10.1023/A:1025722804873. Google Scholar [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. doi: 10.1137/100784497. Google Scholar [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. Google Scholar [16] L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials,, Milan J. Math., 79 (2011), 561. doi: 10.1007/s00032-011-0165-4. Google Scholar [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. doi: 10.3934/dcdsb.2014.19.2013. Google Scholar [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. doi: 10.1016/j.jmaa.2011.08.008. Google Scholar [19] A. Debussche and L. Dettori, On the Cahn-Hilliard equation with a logarithmic free energy,, Nonlinear Anal., 24 (1995), 1491. doi: 10.1016/0362-546X(94)00205-V. Google Scholar [20] C. M. Elliott and S. Zheng, On the Cahn-Hilliard equation,, Arch. Ration. Mech. Anal., 96 (1986), 339. doi: 10.1007/BF00251803. Google Scholar [21] C. M. Elliott and H. Garcke, On the Cahn-Hilliard equation with degenerate mobility,, SIAM J. Math. Anal., 27 (1996), 404. doi: 10.1137/S0036141094267662. Google Scholar [22] P. C. Fife, Models for phase separation and their mathematics,, Electron. J. Differential Equations, 48 (2000). Google Scholar [23] S. Frigeri, C. G. Gal and M. Grasselli, On nonlocal Cahn-Hilliard-Navier-Stokes systems in two dimensions, preprint,, WIAS Preprint, 1923 (2014). Google Scholar [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. doi: 10.1007/s10884-012-9272-3. Google Scholar [25] S. Frigeri and M. Grasselli, Nonlocal Cahn-Hilliard-Navier-Stokes systems with singular potentials,, Dyn. Partial Differ. Equ., 9 (2012), 273. doi: 10.4310/DPDE.2012.v9.n4.a1. Google Scholar [26] H. Gajewski and K. Zacharias, On a nonlocal phase separation model,, J. Math. Anal. Appl., 286 (2003), 11. doi: 10.1016/S0022-247X(02)00425-0. Google Scholar [27] C. G. Gal and M. Grasselli, Longtime behavior of nonlocal Cahn-Hilliard equations,, Discrete Contin. Dyn. Syst., 34 (2014), 145. doi: 10.3934/dcds.2014.34.145. Google Scholar [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. doi: 10.1103/PhysRevLett.76.1094. Google Scholar [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. doi: 10.1007/BF02181479. Google Scholar [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. doi: 10.1137/S0036139996313046. Google Scholar [31] S. C. Glotzer, E. A. Di Marzio and M. Muthukumar, Reaction-controlled morphology of phase separating mixtures,, Phys. Rev. Lett., 74 (1995), 2034. doi: 10.1103/PhysRevLett.74.2034. Google Scholar [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. doi: 10.1002/mats.200300021. Google Scholar [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. doi: 10.1063/1.1571511. Google Scholar [34] T. P. Lodge, Block copolymers: past successes and future challenges,, Macromol. Chem. Phys., 204 (2003), 265. doi: 10.1002/macp.200290073. Google Scholar [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. doi: 10.1016/j.jmaa.2011.02.003. Google Scholar [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. Google Scholar [37] P. Mansky, P. Chaikin and E. L. Thomas, Monolayer films of diblock copolymer microdomains for nanolithographic applications,, J. Mater. Sci., 30 (1995), 1987. doi: 10.1007/BF00353023. Google Scholar [38] S. Melchionna and E. Rocca, On a nonlocal Cahn-Hilliard equation with a reaction term,, Adv. Math. Sci. Appl., 24 (2014), 461. Google Scholar [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. doi: 10.1016/S1874-5717(08)00003-0. Google Scholar [40] A. Miranville, Asymptotic behavior of the Cahn-Hilliard-Oono equation,, J. Appl. Anal. Comput., 1 (2011), 523. Google Scholar [41] A. Miranville, Asymptotic behaviour of a generalized Cahn-Hilliard equation with a proliferation term,, Appl. Anal., 92 (2013), 1308. doi: 10.1080/00036811.2012.671301. Google Scholar [42] C. B. Muratov, Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions,, Comm. Math. Phys., 299 (2010), 45. doi: 10.1007/s00220-010-1094-8. Google Scholar [43] C. B. Muratov, Theory of domain patterns in systems with long-range interactions of Coulomb type,, Phys. Rev. E, 66 (2002). doi: 10.1103/PhysRevE.66.066108. Google Scholar [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. doi: 10.1080/03605308908820597. Google Scholar [45] Y. Nishiura and I. Ohnishi, Some mathematical aspects of the microphase separation in diblock copolymers,, Phys. D, 84 (1995), 31. doi: 10.1016/0167-2789(95)00005-O. Google Scholar [46] A. Novick-Cohen, The Cahn-Hilliard equation: Mathematical and modeling perspectives,, Adv. Math. Sci. Appl., 8 (1998), 965. Google Scholar [47] A. Novick-Cohen, The Cahn-Hilliard equation,, Handbook of differential equations: Evolutionary equations, IV (2008), 201. doi: 10.1016/S1874-5717(08)00004-2. Google Scholar [48] T. Ohta and K. Kawasaki, Equilibrium morphology of block copolymer melts,, Macromolecules, 19 (1986), 2621. doi: 10.1021/ma00164a028. Google Scholar [49] Y. Oono and S. Puri, Computationally Efficient Modeling of Ordering of Quenched Phases,, Phys. Rev. Lett., 58 (1987), 836. doi: 10.1103/PhysRevLett.58.836. Google Scholar [50] G. Schimperna, Global attractors for Cahn-Hilliard equations with nonconstant mobility,, Nonlinearity, 20 (2007), 2365. doi: 10.1088/0951-7715/20/10/006. Google Scholar [51] S. Villain Guillot, 1D Cahn-Hilliard equation for modulated phase systems,, J. Phys. A, 43 (2010). Google Scholar [52] S. Walheim, E. Schaeffer, J. Mlynek and U. Steiner, Nanophase-separated polymer films as high-performance antireflection coatings,, Science, 283 (1999), 520. doi: 10.1126/science.283.5401.520. Google Scholar [53] S. Zheng, Asymptotic behavior of solution to the Cahn-Hilliard equation,, Appl. Anal., 23 (1986), 165. doi: 10.1080/00036818608839639. Google Scholar
 [1] Alain Miranville. Existence of solutions for Cahn-Hilliard type equations. Conference Publications, 2003, 2003 (Special) : 630-637. doi: 10.3934/proc.2003.2003.630 [2] Ciprian G. Gal, Maurizio Grasselli. Longtime behavior of nonlocal Cahn-Hilliard equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 145-179. doi: 10.3934/dcds.2014.34.145 [3] Ahmad Makki, Alain Miranville. Existence of solutions for anisotropic Cahn-Hilliard and Allen-Cahn systems in higher space dimensions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 759-775. doi: 10.3934/dcdss.2016027 [4] Irena Pawłow, Wojciech M. Zajączkowski. Regular weak solutions to 3-D Cahn-Hilliard system in elastic solids. Conference Publications, 2007, 2007 (Special) : 824-833. doi: 10.3934/proc.2007.2007.824 [5] Ciprian G. Gal. On the strong-to-strong interaction case for doubly nonlocal Cahn-Hilliard equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 131-167. doi: 10.3934/dcds.2017006 [6] Álvaro Hernández, Michał Kowalczyk. Rotationally symmetric solutions to the Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 801-827. doi: 10.3934/dcds.2017033 [7] Ciprian G. Gal, Alain Miranville. Robust exponential attractors and convergence to equilibria for non-isothermal Cahn-Hilliard equations with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 113-147. doi: 10.3934/dcdss.2009.2.113 [8] Igor Chueshov, Irena Lasiecka. Existence, uniqueness of weak solutions and global attractors for a class of nonlinear 2D Kirchhoff-Boussinesq models. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 777-809. doi: 10.3934/dcds.2006.15.777 [9] Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127 [10] Annalisa Iuorio, Stefano Melchionna. Long-time behavior of a nonlocal Cahn-Hilliard equation with reaction. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3765-3788. doi: 10.3934/dcds.2018163 [11] Christopher P. Grant. Grain sizes in the discrete Allen-Cahn and Cahn-Hilliard equations. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 127-146. doi: 10.3934/dcds.2001.7.127 [12] Jie Shen, Xiaofeng Yang. Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1669-1691. doi: 10.3934/dcds.2010.28.1669 [13] Alain Miranville, Wafa Saoud, Raafat Talhouk. On the Cahn-Hilliard/Allen-Cahn equations with singular potentials. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3633-3651. doi: 10.3934/dcdsb.2018308 [14] Dimitra Antonopoulou, Georgia Karali. Existence of solution for a generalized stochastic Cahn-Hilliard equation on convex domains. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 31-55. doi: 10.3934/dcdsb.2011.16.31 [15] Peter Howard, Bongsuk Kwon. Spectral analysis for transition front solutions in Cahn-Hilliard systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 125-166. doi: 10.3934/dcds.2012.32.125 [16] Irena Pawłow, Wojciech M. Zajączkowski. On a class of sixth order viscous Cahn-Hilliard type equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 517-546. doi: 10.3934/dcdss.2013.6.517 [17] Tian Ma, Shouhong Wang. Cahn-Hilliard equations and phase transition dynamics for binary systems. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 741-784. doi: 10.3934/dcdsb.2009.11.741 [18] Riccarda Rossi. On two classes of generalized viscous Cahn-Hilliard equations. Communications on Pure & Applied Analysis, 2005, 4 (2) : 405-430. doi: 10.3934/cpaa.2005.4.405 [19] Shihui Zhu. Existence and uniqueness of global weak solutions of the Camassa-Holm equation with a forcing. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5201-5221. doi: 10.3934/dcds.2016026 [20] Changchun Liu, Hui Tang. Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions. Evolution Equations & Control Theory, 2017, 6 (2) : 219-237. doi: 10.3934/eect.2017012

2018 Impact Factor: 1.008