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January  2014, 34(1): 145-179. doi: 10.3934/dcds.2014.34.145

## Longtime behavior of nonlocal Cahn-Hilliard equations

 1 Department of Mathematics, Florida International University, Miami, FL, 33199 2 Dipartimento di Matematica, Politecnico di Milano, 20133 Milano

Received  July 2012 Revised  February 2013 Published  June 2013

Here we consider the nonlocal Cahn-Hilliard equation with constant mobility in a bounded domain. We prove that the associated dynamical system has an exponential attractor, provided that the potential is regular. In order to do that a crucial step is showing the eventual boundedness of the order parameter uniformly with respect to the initial datum. This is obtained through an Alikakos-Moser type argument. We establish a similar result for the viscous nonlocal Cahn-Hilliard equation with singular (e.g., logarithmic) potential. In this case the validity of the so-called separation property is crucial. We also discuss the convergence of a solution to a single stationary state. The separation property in the nonviscous case is known to hold when the mobility degenerates at the pure phases in a proper way and the potential is of logarithmic type. Thus, the existence of an exponential attractor can be proven in this case as well.
Citation: 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
##### References:
 [1] P. W. Bates and F. Chen, Spectral analysis and multidimensional stability of traveling waves for nonlocal Allen-Cahn equation,, J. Math. Anal. Appl., 273 (2002), 45. doi: 10.1016/S0022-247X(02)00205-6. [2] P. W. Bates and A. Chmaj, An integrodifferential model for phase transitions: stationary solutions in higher space dimensions,, J. Statist. Phys., 95 (1999), 1119. doi: 10.1023/A:1004514803625. [3] J. Bedrossian, N. Rodríguez and A. L. Bertozzi, Local and global well-posedness for aggregation equations and Patlak-Keller-Segel models with degenerate diffusion,, Nonlinearity, 24 (2011), 1683. doi: 10.1088/0951-7715/24/6/001. [4] P. W. Bates, P. C. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions,, Arch. Rational Mech. Anal., 138 (1997), 105. doi: 10.1007/s002050050037. [5] 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. [6] 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. [7] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy,, J. Chem. Phys., 28 (1958), 258. doi: 10.1063/1.1744102. [8] X. Chen, Existence, uniqueness and asymptotic stability of traveling waves in nonlocal evolution equations,, Adv. Differential Equations, 2 (1997), 125. [9] 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. [10] 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. [11] P. Colli, P. Krejčí, E. Rocca and J. Sprekels, Nonlinear evolution inclusions arising from phase change models,, Czechoslovak Math. J., 57 (2007), 1067. doi: 10.1007/s10587-007-0114-0. [12] M. Dauge, "Elliptic Boundary Value Problems on Corner Domains. Smoothness and Asymptotics of Solutions,", Lecture Notes in Mathematics, (1341). [13] L. Dung, Remarks on Hölder continuity for parabolic equations and convergence to global attractors,, Nonlinear Analysis, 41 (2000), 921. doi: 10.1016/S0362-546X(98)00319-8. [14] 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. [15] M. Efendiev and S. Zelik, Finite-dimensional attractors and exponential attractors for degenerate doubly nonlinear equations,, Math. Methods Appl. Sci., 32 (2009), 1638. doi: 10.1002/mma.1102. [16] E. Feireisl, F. Issard-Roch and H. Petzeltová, A non-smooth version of the Łojasiewicz-Simon theorem with applications to non-local phase-field systems,, J. Differential Equations, 199 (2004), 1. doi: 10.1016/j.jde.2003.10.026. [17] E. Feireisl and F. Simondon, Convergence for semilinear degenerate parabolic equations in several space dimensions,, J. Dynam. Differential Equations, 12 (2000), 647. doi: 10.1023/A:1026467729263. [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. doi: 10.1007/s10884-012-9272-3. [19] S. Frigeri and M. Grasselli, Nonlocal Cahn-Hilliard-Navier-Stokes systems with singular potentials ,, Dyn. Partial Differ. Equ., 9 (2012), 273. [20] H. Gajewski, On a nonlocal model of non-isothermal phase separation,, Adv. Math. Sci. Appl., 12 (2002), 569. [21] H. Gajewski and K. Gärtner, A dissipative discretization scheme for a nonlocal phase segregation model,, ZAMM Z. Angew. Math. Mech., 85 (2005), 815. doi: 10.1002/zamm.200510233. [22] H. Gajewski and J. A. Griepentrog, A descent method for the free energy of multicomponent systems,, Discrete Contin. Dyn. Syst., 15 (2006), 505. doi: 10.3934/dcds.2006.15.505. [23] 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. [24] C. G. Gal, On a class of degenerate parabolic equations with dynamic boundary conditions,, J. Differential Equations, 253 (2012), 126. doi: 10.1016/j.jde.2012.02.010. [25] C. G. Gal, Sharp estimates for the global attractor of scalar reaction-diffusion equations with a Wentzell boundary condition,, J. Nonlinear Sci., 22 (2012), 85. doi: 10.1007/s00332-011-9109-y. [26] C. G. Gal, Global attractor for a nonlocal model for biological aggregation,, to appear in Comm. Math. Sci., (). [27] C. G. Gal, M. Grasselli and A. Miranville, Robust exponential attractors for singularly perturbed phase-field equations with dynamic boundary conditions,, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 535. doi: 10.1007/s00030-008-7029-9. [28] J. García Melián and J. D. Rossi, A logistic equation with refuge and nonlocal diffusion,, Comm. Pure Appl. Anal., 8 (2009), 2037. doi: 10.3934/cpaa.2009.8.2037. [29] G. Giacomin and J. L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits,, J. Statist. Phys., 87 (1997), 37. doi: 10.1007/BF02181479. [30] G. Giacomin and J. L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions. II. Phase motion,, SIAM J. Appl. Math., 58 (1998), 1707. doi: 10.1137/S0036139996313046. [31] M. Grasselli and G. Schimperna, Nonlocal phase-field systems with general potentials,, Discrete Contin. Dyn. Syst. Ser. A 33 (2013), (2013), 5089. doi: 10.3934/dcds.2013.33.5089. [32] J. Han, The Cauchy problem and steady state solutions for a nonlocal Cahn-Hilliard equation,, Electron. J. Differential Equations, 113 (2004). [33] M. Hassan Farshbaf-Shaker, On a nonlocal viscous phase separation model,, Adv. Math. Sci. Appl., 21 (2011), 187. [34] M. Hassan Farshbaf-Shaker, Existence result for a nonlocal viscous Cahn-Hilliard equation with a degenerate mobility,, preprint, 24 (2011). [35] 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. doi: 10.3934/dcdss.2011.4.653. [36] 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. [37] A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials,, Math. Meth. Appl. Sci., 27 (2004), 545. doi: 10.1002/mma.464. [38] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, in, (2008), 103. doi: 10.1016/S1874-5717(08)00003-0. [39] A. Novick-Cohen, On the viscous Cahn-Hilliard equation,, in, (1988), 1985. [40] A. Novick-Cohen, The Cahn-Hilliard equation,, in, (2008), 201. doi: 10.1016/S1874-5717(08)00004-2. [41] J. S. Rowlinson, Translation of J. D. van der Waals, The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density,, J. Statist. Phys., 20 (1979), 197. doi: 10.1007/BF01011513.

show all references

##### References:
 [1] P. W. Bates and F. Chen, Spectral analysis and multidimensional stability of traveling waves for nonlocal Allen-Cahn equation,, J. Math. Anal. Appl., 273 (2002), 45. doi: 10.1016/S0022-247X(02)00205-6. [2] P. W. Bates and A. Chmaj, An integrodifferential model for phase transitions: stationary solutions in higher space dimensions,, J. Statist. Phys., 95 (1999), 1119. doi: 10.1023/A:1004514803625. [3] J. Bedrossian, N. Rodríguez and A. L. Bertozzi, Local and global well-posedness for aggregation equations and Patlak-Keller-Segel models with degenerate diffusion,, Nonlinearity, 24 (2011), 1683. doi: 10.1088/0951-7715/24/6/001. [4] P. W. Bates, P. C. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions,, Arch. Rational Mech. Anal., 138 (1997), 105. doi: 10.1007/s002050050037. [5] 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. [6] 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. [7] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy,, J. Chem. Phys., 28 (1958), 258. doi: 10.1063/1.1744102. [8] X. Chen, Existence, uniqueness and asymptotic stability of traveling waves in nonlocal evolution equations,, Adv. Differential Equations, 2 (1997), 125. [9] 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. [10] 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. [11] P. Colli, P. Krejčí, E. Rocca and J. Sprekels, Nonlinear evolution inclusions arising from phase change models,, Czechoslovak Math. J., 57 (2007), 1067. doi: 10.1007/s10587-007-0114-0. [12] M. Dauge, "Elliptic Boundary Value Problems on Corner Domains. Smoothness and Asymptotics of Solutions,", Lecture Notes in Mathematics, (1341). [13] L. Dung, Remarks on Hölder continuity for parabolic equations and convergence to global attractors,, Nonlinear Analysis, 41 (2000), 921. doi: 10.1016/S0362-546X(98)00319-8. [14] 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. [15] M. Efendiev and S. Zelik, Finite-dimensional attractors and exponential attractors for degenerate doubly nonlinear equations,, Math. Methods Appl. Sci., 32 (2009), 1638. doi: 10.1002/mma.1102. [16] E. Feireisl, F. Issard-Roch and H. Petzeltová, A non-smooth version of the Łojasiewicz-Simon theorem with applications to non-local phase-field systems,, J. Differential Equations, 199 (2004), 1. doi: 10.1016/j.jde.2003.10.026. [17] E. Feireisl and F. Simondon, Convergence for semilinear degenerate parabolic equations in several space dimensions,, J. Dynam. Differential Equations, 12 (2000), 647. doi: 10.1023/A:1026467729263. [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. doi: 10.1007/s10884-012-9272-3. [19] S. Frigeri and M. Grasselli, Nonlocal Cahn-Hilliard-Navier-Stokes systems with singular potentials ,, Dyn. Partial Differ. Equ., 9 (2012), 273. [20] H. Gajewski, On a nonlocal model of non-isothermal phase separation,, Adv. Math. Sci. Appl., 12 (2002), 569. [21] H. Gajewski and K. Gärtner, A dissipative discretization scheme for a nonlocal phase segregation model,, ZAMM Z. Angew. Math. Mech., 85 (2005), 815. doi: 10.1002/zamm.200510233. [22] H. Gajewski and J. A. Griepentrog, A descent method for the free energy of multicomponent systems,, Discrete Contin. Dyn. Syst., 15 (2006), 505. doi: 10.3934/dcds.2006.15.505. [23] 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. [24] C. G. Gal, On a class of degenerate parabolic equations with dynamic boundary conditions,, J. Differential Equations, 253 (2012), 126. doi: 10.1016/j.jde.2012.02.010. [25] C. G. Gal, Sharp estimates for the global attractor of scalar reaction-diffusion equations with a Wentzell boundary condition,, J. Nonlinear Sci., 22 (2012), 85. doi: 10.1007/s00332-011-9109-y. [26] C. G. Gal, Global attractor for a nonlocal model for biological aggregation,, to appear in Comm. Math. Sci., (). [27] C. G. Gal, M. Grasselli and A. Miranville, Robust exponential attractors for singularly perturbed phase-field equations with dynamic boundary conditions,, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 535. doi: 10.1007/s00030-008-7029-9. [28] J. García Melián and J. D. Rossi, A logistic equation with refuge and nonlocal diffusion,, Comm. Pure Appl. Anal., 8 (2009), 2037. doi: 10.3934/cpaa.2009.8.2037. [29] G. Giacomin and J. L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits,, J. Statist. Phys., 87 (1997), 37. doi: 10.1007/BF02181479. [30] G. Giacomin and J. L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions. II. Phase motion,, SIAM J. Appl. Math., 58 (1998), 1707. doi: 10.1137/S0036139996313046. [31] M. Grasselli and G. Schimperna, Nonlocal phase-field systems with general potentials,, Discrete Contin. Dyn. Syst. Ser. A 33 (2013), (2013), 5089. doi: 10.3934/dcds.2013.33.5089. [32] J. Han, The Cauchy problem and steady state solutions for a nonlocal Cahn-Hilliard equation,, Electron. J. Differential Equations, 113 (2004). [33] M. Hassan Farshbaf-Shaker, On a nonlocal viscous phase separation model,, Adv. Math. Sci. Appl., 21 (2011), 187. [34] M. Hassan Farshbaf-Shaker, Existence result for a nonlocal viscous Cahn-Hilliard equation with a degenerate mobility,, preprint, 24 (2011). [35] 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. doi: 10.3934/dcdss.2011.4.653. [36] 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. [37] A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials,, Math. Meth. Appl. Sci., 27 (2004), 545. doi: 10.1002/mma.464. [38] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, in, (2008), 103. doi: 10.1016/S1874-5717(08)00003-0. [39] A. Novick-Cohen, On the viscous Cahn-Hilliard equation,, in, (1988), 1985. [40] A. Novick-Cohen, The Cahn-Hilliard equation,, in, (2008), 201. doi: 10.1016/S1874-5717(08)00004-2. [41] J. S. Rowlinson, Translation of J. D. van der Waals, The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density,, J. Statist. Phys., 20 (1979), 197. doi: 10.1007/BF01011513.
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