July  2016, 15(4): 1077-1105. doi: 10.3934/cpaa.2016.15.1077

Large time behavior of a conserved phase-field system

1. 

Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, P.O. Box 546, Dhahran 31261, Saudi Arabia, Saudi Arabia

Received  December 2015 Revised  February 2016 Published  April 2016

We investigate the large time behavior of a conserved phase-field system that describes the phase separation in a material with viscosity effects. We prove a well-posedness result, the existence of the global attractor and its upper semicontinuity, when the heat capacity tends to zero. Then we prove the existence of inertial manifolds in one space dimension, and for the case of a rectangular domain in two space dimension. We also construct robust families of exponential attractors that converge in the sense of upper and lower semicontinuity to those of the viscous Cahn-Hilliard equation. Continuity properties of the intersection of the inertial manifolds with bounded absorbing sets are also proven. This work extends and improves some recent results proven by A. Bonfoh for both the conserved and non-conserved phase-field systems.
Citation: Ahmed Bonfoh, Cyril D. Enyi. Large time behavior of a conserved phase-field system. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1077-1105. doi: 10.3934/cpaa.2016.15.1077
References:
[1]

A. Bonfoh, Dynamics of the conserved phase-field system,, \emph{Appl. Analysis}, 95 (2016), 44. doi: 10.1080/00036811.2014.997225. Google Scholar

[2]

A. Bonfoh, The singular limit dynamics of the phase-field equations,, \emph{Ann. Mat. Pura Appl.}, 190 (2011), 105. doi: 10.1007/s10231-010-0141-6. Google Scholar

[3]

A. Bonfoh, M. Grasselli and A. Miranville, Singularly perturbed 1D Cahn-Hilliard equation revisited,, \emph{Nonlinear Differ. Equ. Appli.}, 17 (2010), 663. doi: 10.1007/s00030-010-0075-0. Google Scholar

[4]

A. Bonfoh and A. Miranville, On Cahn-Hilliard-Gurtin equations,, \emph{Nonlinear Anal.}, 47 (2001), 3455. doi: 10.1016/S0362-546X(01)00463-1. Google Scholar

[5]

D. Brochet, Maximal attractor and inertial sets for some second and fourth order phase field models,, \emph{Pitman Res. Notes Math. Ser.}, 296 (1993), 77. Google Scholar

[6]

D. Brochet, D. Hilhorst and A. Novick-Cohen, Maximal attractor and inertial sets for a conserved phase field model,, \emph{Adv. Diff. Eqns}, 1 (1996), 547. Google Scholar

[7]

G. Caginalp, Conserved-phase field system: implications for kinetic undercooling,, \emph{Phys. Rev. B}, 38 (1988), 789. Google Scholar

[8]

S.-N. Chow and K. Lu, Invariant manifolds for flow in Banach spaces,, \emph{J. Diff. Eqns}, 74 (1988), 285. doi: 10.1016/0022-0396(88)90007-1. Google Scholar

[9]

S.-N. Chow, K. Lu and G.R. Sell, Smoothness of inertial manifolds,, \emph{J. Math. Anal. Appl.}, 169 (1992), 283. doi: 10.1016/0022-247X(92)90115-T. Google Scholar

[10]

C.M. Elliott and A.M. Stuart, The viscous Cahn-Hilliard equation. II. Analysis,, \emph{J. Differential Equations}, 128 (1996), 387. doi: 10.1006/jdeq.1996.0101. Google Scholar

[11]

S. Gatti, M. Grasselli, A. Miranville and V. Pata, A construction of a robust family of exponential attractors,, \emph{Proc. Amer. Math. Soc.}, 134 (2006), 117. doi: 10.1090/S0002-9939-05-08340-1. Google Scholar

[12]

S. Gatti, M. Grasselli, A. Miranville and V. Pata, Hyperbolic relaxation of the viscous Cahn-Hilliard equation in 3-D,, \emph{Math. Models Methods Appl. Sciences}, 15 (2005), 165. doi: 10.1142/S0218202505000327. Google Scholar

[13]

G. Gilardi, On a conserved phase field model with irregular potentials and dynamic boundary conditions,, \emph{Istit. Lombardo Accad. Sci. Lett. Rend. A}, 141 (2007), 129. Google Scholar

[14]

A. Miranville, On the conserved phase-field model,, \emph{J. Math. Anal. Appl.}, 400 (2013), 143. doi: 10.1016/j.jmaa.2012.11.038. Google Scholar

[15]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, Evolutionary equations. Vol. IV, (2008), 103. doi: 10.1016/S1874-5717(08)00003-0. Google Scholar

[16]

G. Mola, Global attractors for a three-dimensional conserved phase-field system with memory,, \emph{Comm. Pure Appl. Anal.}, 7 (2008), 317. doi: 10.3934/cpaa.2008.7.317. Google Scholar

[17]

G. Mola, Stability of global and exponential attractors for a three-dimensional conserved phase-field system with memory,, \emph{Math. Models Methods Appl. Sciences}, 32 (2009), 2368. doi: 10.1002/mma.1139. Google Scholar

[18]

X. Mora and J. Solà-Morales, The singular limit dynamics of semilineardamped wave equations,, \emph{J. Differential Equations}, 78 (1989), 262. doi: 10.1016/0022-0396(89)90065-X. Google Scholar

[19]

B. Nicolaenko, B. Scheurer and R. Temam, Some global dynamical properties of a class of pattern formation equations,, \emph{Comm. Partial Differential Equations}, 14 (1989), 245. doi: 10.1080/03605308908820597. Google Scholar

[20]

A. Novick-Cohen, On the viscous Cahn-Hilliard equation,, in \emph{Material Instabilities in Continuum Mechanics} (Edinburgh, (1988), 1985. Google Scholar

[21]

I. Richards, On the gaps between numbers which are sums of two squares,, \emph{Adv. Math.}, 46 (1982), 1. doi: 10.1016/0001-8708(82)90051-2. Google Scholar

[22]

J.C. Robinson, Infinite-dimensional Dynamical Systems. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors,, Cambridge Texts in Applied Mathematics, (2001). doi: 10.1007/978-94-010-0732-0. Google Scholar

[23]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, 2nd Edition, (1997). doi: 10.1007/978-1-4612-0645-3. Google Scholar

[24]

S. Zheng and A. Milani, Exponential attractors and inertial manifolds for singular perturbations of the Cahn-Hilliard equations,, \emph{Nonlinear Anal.}, 57 (2004), 843. doi: 10.1016/j.na.2004.03.023. Google Scholar

show all references

References:
[1]

A. Bonfoh, Dynamics of the conserved phase-field system,, \emph{Appl. Analysis}, 95 (2016), 44. doi: 10.1080/00036811.2014.997225. Google Scholar

[2]

A. Bonfoh, The singular limit dynamics of the phase-field equations,, \emph{Ann. Mat. Pura Appl.}, 190 (2011), 105. doi: 10.1007/s10231-010-0141-6. Google Scholar

[3]

A. Bonfoh, M. Grasselli and A. Miranville, Singularly perturbed 1D Cahn-Hilliard equation revisited,, \emph{Nonlinear Differ. Equ. Appli.}, 17 (2010), 663. doi: 10.1007/s00030-010-0075-0. Google Scholar

[4]

A. Bonfoh and A. Miranville, On Cahn-Hilliard-Gurtin equations,, \emph{Nonlinear Anal.}, 47 (2001), 3455. doi: 10.1016/S0362-546X(01)00463-1. Google Scholar

[5]

D. Brochet, Maximal attractor and inertial sets for some second and fourth order phase field models,, \emph{Pitman Res. Notes Math. Ser.}, 296 (1993), 77. Google Scholar

[6]

D. Brochet, D. Hilhorst and A. Novick-Cohen, Maximal attractor and inertial sets for a conserved phase field model,, \emph{Adv. Diff. Eqns}, 1 (1996), 547. Google Scholar

[7]

G. Caginalp, Conserved-phase field system: implications for kinetic undercooling,, \emph{Phys. Rev. B}, 38 (1988), 789. Google Scholar

[8]

S.-N. Chow and K. Lu, Invariant manifolds for flow in Banach spaces,, \emph{J. Diff. Eqns}, 74 (1988), 285. doi: 10.1016/0022-0396(88)90007-1. Google Scholar

[9]

S.-N. Chow, K. Lu and G.R. Sell, Smoothness of inertial manifolds,, \emph{J. Math. Anal. Appl.}, 169 (1992), 283. doi: 10.1016/0022-247X(92)90115-T. Google Scholar

[10]

C.M. Elliott and A.M. Stuart, The viscous Cahn-Hilliard equation. II. Analysis,, \emph{J. Differential Equations}, 128 (1996), 387. doi: 10.1006/jdeq.1996.0101. Google Scholar

[11]

S. Gatti, M. Grasselli, A. Miranville and V. Pata, A construction of a robust family of exponential attractors,, \emph{Proc. Amer. Math. Soc.}, 134 (2006), 117. doi: 10.1090/S0002-9939-05-08340-1. Google Scholar

[12]

S. Gatti, M. Grasselli, A. Miranville and V. Pata, Hyperbolic relaxation of the viscous Cahn-Hilliard equation in 3-D,, \emph{Math. Models Methods Appl. Sciences}, 15 (2005), 165. doi: 10.1142/S0218202505000327. Google Scholar

[13]

G. Gilardi, On a conserved phase field model with irregular potentials and dynamic boundary conditions,, \emph{Istit. Lombardo Accad. Sci. Lett. Rend. A}, 141 (2007), 129. Google Scholar

[14]

A. Miranville, On the conserved phase-field model,, \emph{J. Math. Anal. Appl.}, 400 (2013), 143. doi: 10.1016/j.jmaa.2012.11.038. Google Scholar

[15]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, Evolutionary equations. Vol. IV, (2008), 103. doi: 10.1016/S1874-5717(08)00003-0. Google Scholar

[16]

G. Mola, Global attractors for a three-dimensional conserved phase-field system with memory,, \emph{Comm. Pure Appl. Anal.}, 7 (2008), 317. doi: 10.3934/cpaa.2008.7.317. Google Scholar

[17]

G. Mola, Stability of global and exponential attractors for a three-dimensional conserved phase-field system with memory,, \emph{Math. Models Methods Appl. Sciences}, 32 (2009), 2368. doi: 10.1002/mma.1139. Google Scholar

[18]

X. Mora and J. Solà-Morales, The singular limit dynamics of semilineardamped wave equations,, \emph{J. Differential Equations}, 78 (1989), 262. doi: 10.1016/0022-0396(89)90065-X. Google Scholar

[19]

B. Nicolaenko, B. Scheurer and R. Temam, Some global dynamical properties of a class of pattern formation equations,, \emph{Comm. Partial Differential Equations}, 14 (1989), 245. doi: 10.1080/03605308908820597. Google Scholar

[20]

A. Novick-Cohen, On the viscous Cahn-Hilliard equation,, in \emph{Material Instabilities in Continuum Mechanics} (Edinburgh, (1988), 1985. Google Scholar

[21]

I. Richards, On the gaps between numbers which are sums of two squares,, \emph{Adv. Math.}, 46 (1982), 1. doi: 10.1016/0001-8708(82)90051-2. Google Scholar

[22]

J.C. Robinson, Infinite-dimensional Dynamical Systems. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors,, Cambridge Texts in Applied Mathematics, (2001). doi: 10.1007/978-94-010-0732-0. Google Scholar

[23]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, 2nd Edition, (1997). doi: 10.1007/978-1-4612-0645-3. Google Scholar

[24]

S. Zheng and A. Milani, Exponential attractors and inertial manifolds for singular perturbations of the Cahn-Hilliard equations,, \emph{Nonlinear Anal.}, 57 (2004), 843. doi: 10.1016/j.na.2004.03.023. Google Scholar

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