Longtime behavior of nonlocal Cahn-Hilliard equations

Ciprian G. Gal; Maurizio Grasselli

doi:10.3934/dcds.2014.34.145

Article Contents

2014, Volume 34, Issue 1: 145-179. Doi: 10.3934/dcds.2014.34.145

This issue Previous Article Dissipative solutions and the incompressible inviscid limits of the compressible magnetohydrodynamic system in unbounded domains Next Article Regularity of pullback attractors and attraction in $H^1$ in arbitrarily large finite intervals for 2D Navier-Stokes equations with infinite delay

Longtime behavior of nonlocal Cahn-Hilliard equations

Ciprian G. Gal^1, and
Maurizio Grasselli^2,

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 online: June 2013

Abstract / Introduction Related Papers Cited by

Abstract

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.

Keywords:

Cahn-Hilliard equation,
nonlocal interactions,
regular and singular potentials,
degenerate mobility,
well-posedness,
uniform boundedness of solutions,
exponential attractors,
convergence to single stationary states.

Mathematics Subject Classification: 35R09, 37L30, 82C24.

Citation:

Related Papers

Cited by

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), 45-57.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), 1119-1139.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), 1683-1714.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), 105-136.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), 235-277.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), 289-312.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), 258-267.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), 125-160.
[9]	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.
[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), 428-444.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), 1067-1098.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, Vol. 1341, Springer-Verlag, Berlin, 1988.
[13]	L. Dung, Remarks on Hölder continuity for parabolic equations and convergence to global attractors, Nonlinear Analysis, 41 (2000), 921-941.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), 1491-1514.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), 1638-1668.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), 1-21.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), 647-673.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), 827-856.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), 273-304.
[20]	H. Gajewski, On a nonlocal model of non-isothermal phase separation, Adv. Math. Sci. Appl., 12 (2002), 569-586.
[21]	H. Gajewski and K. Gärtner, A dissipative discretization scheme for a nonlocal phase segregation model, ZAMM Z. Angew. Math. Mech., 85 (2005), 815-822.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), 505-528.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), 11-31.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), 126-166.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), 85-106.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., arXiv:1302.5351.
[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), 535-556.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), 2037-2053.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), 37-61.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), 1707-1729.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), 5089-5106.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), 9 pp.
[33]	M. Hassan Farshbaf-Shaker, On a nonlocal viscous phase separation model, Adv. Math. Sci. Appl., 21 (2011), 187-222.
[34]	M. Hassan Farshbaf-Shaker, Existence result for a nonlocal viscous Cahn-Hilliard equation with a degenerate mobility, preprint, Universität Regensburg, 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), 653-670.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), 724-735.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), 545-582.doi: 10.1002/mma.464.
[38]	A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in "Handbook of Differential Equations: Evolutionary Equations," Vol. IV, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, (2008), 103-200.doi: 10.1016/S1874-5717(08)00003-0.
[39]	A. Novick-Cohen, On the viscous Cahn-Hilliard equation, in "Material instabilities in continuum mechanics" (Edinburgh, 1985-1986), Oxford Sci. Publ., Oxford Univ. Press, New York, (1988), 329-342.
[40]	A. Novick-Cohen, The Cahn-Hilliard equation, in "Evolutionary Equations," Vol. IV, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, (2008), 201-228.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), 197-244.doi: 10.1007/BF01011513.