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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

X. Chen, Existence, uniqueness and asymptotic stability of traveling waves in nonlocal evolution equations,, Adv. Differential Equations, 2 (1997), 125. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

M. Dauge, "Elliptic Boundary Value Problems on Corner Domains. Smoothness and Asymptotics of Solutions,", Lecture Notes in Mathematics, (1341). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

S. Frigeri and M. Grasselli, Nonlocal Cahn-Hilliard-Navier-Stokes systems with singular potentials ,, Dyn. Partial Differ. Equ., 9 (2012), 273. Google Scholar

[20]

H. Gajewski, On a nonlocal model of non-isothermal phase separation,, Adv. Math. Sci. Appl., 12 (2002), 569. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

C. G. Gal, Global attractor for a nonlocal model for biological aggregation,, to appear in Comm. Math. Sci., (). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[32]

J. Han, The Cauchy problem and steady state solutions for a nonlocal Cahn-Hilliard equation,, Electron. J. Differential Equations, 113 (2004). Google Scholar

[33]

M. Hassan Farshbaf-Shaker, On a nonlocal viscous phase separation model,, Adv. Math. Sci. Appl., 21 (2011), 187. Google Scholar

[34]

M. Hassan Farshbaf-Shaker, Existence result for a nonlocal viscous Cahn-Hilliard equation with a degenerate mobility,, preprint, 24 (2011). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[39]

A. Novick-Cohen, On the viscous Cahn-Hilliard equation,, in, (1988), 1985. Google Scholar

[40]

A. Novick-Cohen, The Cahn-Hilliard equation,, in, (2008), 201. doi: 10.1016/S1874-5717(08)00004-2. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

X. Chen, Existence, uniqueness and asymptotic stability of traveling waves in nonlocal evolution equations,, Adv. Differential Equations, 2 (1997), 125. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

M. Dauge, "Elliptic Boundary Value Problems on Corner Domains. Smoothness and Asymptotics of Solutions,", Lecture Notes in Mathematics, (1341). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

S. Frigeri and M. Grasselli, Nonlocal Cahn-Hilliard-Navier-Stokes systems with singular potentials ,, Dyn. Partial Differ. Equ., 9 (2012), 273. Google Scholar

[20]

H. Gajewski, On a nonlocal model of non-isothermal phase separation,, Adv. Math. Sci. Appl., 12 (2002), 569. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

C. G. Gal, Global attractor for a nonlocal model for biological aggregation,, to appear in Comm. Math. Sci., (). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[32]

J. Han, The Cauchy problem and steady state solutions for a nonlocal Cahn-Hilliard equation,, Electron. J. Differential Equations, 113 (2004). Google Scholar

[33]

M. Hassan Farshbaf-Shaker, On a nonlocal viscous phase separation model,, Adv. Math. Sci. Appl., 21 (2011), 187. Google Scholar

[34]

M. Hassan Farshbaf-Shaker, Existence result for a nonlocal viscous Cahn-Hilliard equation with a degenerate mobility,, preprint, 24 (2011). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[39]

A. Novick-Cohen, On the viscous Cahn-Hilliard equation,, in, (1988), 1985. Google Scholar

[40]

A. Novick-Cohen, The Cahn-Hilliard equation,, in, (2008), 201. doi: 10.1016/S1874-5717(08)00004-2. Google Scholar

[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. Google Scholar

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