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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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