# American Institute of Mathematical Sciences

October  2021, 20(10): 3655-3682. doi: 10.3934/cpaa.2021125

## Robust exponential attractors for singularly perturbed conserved phase-field systems with no growth assumption on the nonlinear term

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

* Corresponding author

Received  December 2020 Revised  June 2021 Published  October 2021 Early access  July 2021

We consider the conserved phase-field system
 $\left\{ \begin{array}{l}\tau {\phi _t} + N(\delta {\phi _t} + N\phi + g(\phi ) - u) = 0,\\\epsilon{u_t} + {\phi _t} + Nu = 0,\end{array} \right.\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {{{\rm{S}}_\varepsilon }} \right)$
where
 $\tau>0$
is a relaxation time,
 $\delta>0$
is the viscosity parameter,
 $\epsilon\in (0,1]$
is the heat capacity,
 $\phi$
is the order parameter,
 $u$
is the absolute temperature, the Laplace operator
 $N = -\Delta:{\mathscr D}(N)\to \dot L^2(\Omega)$
is subject to either Neumann boundary conditions (in which case
 $\Omega\subset{\mathbb R}^d$
is a bounded domain with smooth boundary) or periodic boundary conditions (in which case
 $\Omega = \Pi_{i = 1}^d(0,L_i),$
 $L_i>0$
),
 $d = 1,2$
or 3, and
 $G(\phi) = \int_0^\phi g(\sigma)d\sigma$
is a double-well potential. Let
 $j = 1$
when
 $d = 1$
and
 $j = 2$
when
 $d = 2$
or 3. We assume that
 $g\in{\mathcal C}^{j+1}(\mathbb R)$
and satisfies the conditions
 $g'(\phi)\geq -{\mathscr C}_1$
,
 $G(\phi)\ge -{\mathscr C}_2$
and
 $(\phi-m(\phi))g(\phi)-{\mathscr C}_3(m(\phi))G(s)\ge -{\mathscr C}_4(m(\phi))$
(
 ${\mathscr C}_5(\varrho)\le {\mathscr C}_l(m(\phi))\le {\mathscr C}_6(\varrho)$
,
 $l = 3,4$
, whenever
 $|m(\phi)|\le \varrho$
), where
 $\varrho,{\mathscr C}_1, {\mathscr C}_2,{\mathscr C}_4\ge 0$
,
 ${\mathscr C}_3, {\mathscr C}_5,{\mathscr C}_6>0$
and
 $m(\phi) = \frac{1}{|\Omega|}\int_\Omega\phi(x)dx$
. For instance,
 $g(\phi) = \sum_{k = 1}^{2p-1}a_k\phi^k,$
 $p\in{\mathbb N},$
 $p\ge 2,$
 $a_{2p-1}>0,$
satisfies all the above-mentioned conditions. We then prove a well-posedness result, the existence of the global attractor and a family of exponential attractors in the phase space
 ${\mathcal V}_j = {\mathscr D}(N^{j/2})\times{\mathscr D}(N^{j/2})$
equipped with the norm
 $\|(\psi,\varphi)\|_{{\mathcal V}_{j}} = (\|N^{j/2}\psi\|^2+m(\psi)^2+\|N^{j/2}\varphi\|^2+m(\varphi)^2)^{1/2}$
. Moreover, we demonstrate that the global attractor is upper semicontinuous at
 $\epsilon = 0$
in the metric induced by the norm
 $\|.\|_{{\mathcal V}_{j+1}}$
. In addition, the exponential attractors are proven to be Hölder continuous at
 $\epsilon = 0$
in the metric induced by the norm
 $\|.\|_{{\mathcal V}_{j}}$
. Our results improve a recent work by Bonfoh and Enyi [Comm. Pure Appl. Anal. 2016; 35:1077-1105] where the following additional growth condition
 $|g''(\phi)|\leq {\mathscr C}_7\left(|\phi|^{p}+1\right),$
 ${\mathscr C}_7>0$
,
 $p>0$
is arbitrary when
 $d = 1, 2$
and
 $p\in [0,3]$
when
 $d = 3$
, was required, preventing
 $g$
to be a polynomial of any arbitrary odd degree with a strictly positive leading coefficient in three space dimension.
Citation: Ahmed Bonfoh, Ibrahim A. Suleman. Robust exponential attractors for singularly perturbed conserved phase-field systems with no growth assumption on the nonlinear term. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3655-3682. doi: 10.3934/cpaa.2021125
##### References:
 [1] A. Babin and B. Nicolaenko, Exponential attractors of reaction-diffusion systems in an unbounded domain, J. Dyn. Differ. Equ., 7 (1995), 567-589.  doi: 10.1007/BF02218725. [2] A. Bonfoh, Dynamics of the conserved phase-field system, Appl. Anal., 95 (2016), 44-62.  doi: 10.1080/00036811.2014.997225. [3] A. Bonfoh and C. D. Enyi, Large time behavior of a conserved phase-field system, Comm. Pure Appl. Anal., 15 (2016), 1077-1105.  doi: 10.3934/cpaa.2016.15.1077. [4] A. Bonfoh and C. D. Enyi, The Cahn-Hilliard equation as limit of a conserved phase-field system, Asymptotic Anal., 101 (2017), 97-148.  doi: 10.3233/ASY-161395. [5] D. Brochet, Maximal attractor and inertial sets for some second and fourth order phase field models, Pitman Res. Notes Math. Ser., vol. 296, Longman Sci. Tech., Harlow, 1993, 77–85. [6] D. Brochet, D. Hilhorst and A. Novick-Cohen, Maximal attractor and inertial sets for a conserved phase field model, Adv. Diff. Eqns, 1 (1996), 547-568. [7] G. Caginalp, Conserved-phase field system: implications for kinetic undercooling, Phys. Rev. B, 38 (1988), 789-791. [8] L. Dung and B. Nicolaenko, Exponential attractors in Banach spaces, J. Dyn. Differ. Equ., 13 (2001), 791-806.  doi: 10.1023/A:1016676027666. [9] A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, Masson, Paris, 1994. [10] M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\mathbb R^3$, C. R. Math. Acad. Sci. Paris, 330 (2000), 713-718.  doi: 10.1016/S0764-4442(00)00259-7. [11] M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nachr., 272 (2004), 11-31.  doi: 10.1002/mana.200310186. [12] C. M. Elliott and A. M. Stuart, The viscous Cahn-Hilliard equation. Ⅱ. Analysis, J. Differ. Equ., 128 (1996), 387-414.  doi: 10.1006/jdeq.1996.0101. [13] S. Gatti, M. Grasselli, A. Miranville and V. Pata, A construction of a robust family of exponential attractors, Proc. Amer. Math. Soc., 134 (2006), 117-127.  doi: 10.1090/S0002-9939-05-08340-1. [14] S. Gatti, M. Grasselli, A. Miranville and V. Pata, Hyperbolic relaxation of the viscous Cahn-Hilliard equation in 3-D, Math. Models Methods Appl. Sci., 15 (2005), 165-198.  doi: 10.1142/S0218202505000327. [15] G. Gilardi, On a conserved phase field model with irregular potentials and dynamic boundary conditions, Istit. Lombardo Accad. Sci. Lett. Rend. A, 141 (2007), 129-161. [16] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988. doi: 10.1090/surv/025. [17] J. K. Hale and G. Raugel, Upper-semicontinuity of the attractor for a singularly perturbed hyperbolic equation, J. Differ. Equ., 73 (1988), 197-214.  doi: 10.1016/0022-0396(88)90104-0. [18] J. K. Hale and G. Raugel, Lower-semicontinuity of the attractor for a singularly perturbed hyperbolic equation, J. Dyn. Differ. Equ., 2 (1990), 19-67.  doi: 10.1007/BF01047769. [19] A. Miranville, Exponential attractors for a class of evolution equations by a decomposition method, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 145-150.  doi: 10.1016/S0764-4442(99)80153-0. [20] A. Miranville, On the conserved phase-field model, J. Math. Anal. Appl., 400 (2013), 143-152.  doi: 10.1016/j.jmaa.2012.11.038. [21] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, Handb. Differ. Equ., 4 (2018), 103-200. doi: 10.1016/S1874-5717(08)00003-0. [22] G. Mola, Global attractors for a three-dimensional conserved phase-field system with memory, Commun. Pure Appl. Anal., 7 (2008), 317-353.  doi: 10.3934/cpaa.2008.7.317. [23] G. Mola, Stability of global and exponential attractors for a three-dimensional conserved phase-field system with memory, Math. Models Methods Appl. Sci., 32 (2009), 2368-2404.  doi: 10.1002/mma.1139. [24] A. Novick-Cohen, On the viscous Cahn-Hilliard equation, in Material Instabilities in Continuum Mechanics, Oxford Univ. Press, New York, 1988. [25] G. Raugel, Singularly perturbed hyperbolic equations revisited, in International Conference on Differential Equations, World Sci. Publishing, River Edge, NJ, 2000. [26] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd Edition, Springer-Verlag, Berlin, Heidelberg, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

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##### References:
 [1] A. Babin and B. Nicolaenko, Exponential attractors of reaction-diffusion systems in an unbounded domain, J. Dyn. Differ. Equ., 7 (1995), 567-589.  doi: 10.1007/BF02218725. [2] A. Bonfoh, Dynamics of the conserved phase-field system, Appl. Anal., 95 (2016), 44-62.  doi: 10.1080/00036811.2014.997225. [3] A. Bonfoh and C. D. Enyi, Large time behavior of a conserved phase-field system, Comm. Pure Appl. Anal., 15 (2016), 1077-1105.  doi: 10.3934/cpaa.2016.15.1077. [4] A. Bonfoh and C. D. Enyi, The Cahn-Hilliard equation as limit of a conserved phase-field system, Asymptotic Anal., 101 (2017), 97-148.  doi: 10.3233/ASY-161395. [5] D. Brochet, Maximal attractor and inertial sets for some second and fourth order phase field models, Pitman Res. Notes Math. Ser., vol. 296, Longman Sci. Tech., Harlow, 1993, 77–85. [6] D. Brochet, D. Hilhorst and A. Novick-Cohen, Maximal attractor and inertial sets for a conserved phase field model, Adv. Diff. Eqns, 1 (1996), 547-568. [7] G. Caginalp, Conserved-phase field system: implications for kinetic undercooling, Phys. Rev. B, 38 (1988), 789-791. [8] L. Dung and B. Nicolaenko, Exponential attractors in Banach spaces, J. Dyn. Differ. Equ., 13 (2001), 791-806.  doi: 10.1023/A:1016676027666. [9] A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, Masson, Paris, 1994. [10] M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\mathbb R^3$, C. R. Math. Acad. Sci. Paris, 330 (2000), 713-718.  doi: 10.1016/S0764-4442(00)00259-7. [11] M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nachr., 272 (2004), 11-31.  doi: 10.1002/mana.200310186. [12] C. M. Elliott and A. M. Stuart, The viscous Cahn-Hilliard equation. Ⅱ. Analysis, J. Differ. Equ., 128 (1996), 387-414.  doi: 10.1006/jdeq.1996.0101. [13] S. Gatti, M. Grasselli, A. Miranville and V. Pata, A construction of a robust family of exponential attractors, Proc. Amer. Math. Soc., 134 (2006), 117-127.  doi: 10.1090/S0002-9939-05-08340-1. [14] S. Gatti, M. Grasselli, A. Miranville and V. Pata, Hyperbolic relaxation of the viscous Cahn-Hilliard equation in 3-D, Math. Models Methods Appl. Sci., 15 (2005), 165-198.  doi: 10.1142/S0218202505000327. [15] G. Gilardi, On a conserved phase field model with irregular potentials and dynamic boundary conditions, Istit. Lombardo Accad. Sci. Lett. Rend. A, 141 (2007), 129-161. [16] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988. doi: 10.1090/surv/025. [17] J. K. Hale and G. Raugel, Upper-semicontinuity of the attractor for a singularly perturbed hyperbolic equation, J. Differ. Equ., 73 (1988), 197-214.  doi: 10.1016/0022-0396(88)90104-0. [18] J. K. Hale and G. Raugel, Lower-semicontinuity of the attractor for a singularly perturbed hyperbolic equation, J. Dyn. Differ. Equ., 2 (1990), 19-67.  doi: 10.1007/BF01047769. [19] A. Miranville, Exponential attractors for a class of evolution equations by a decomposition method, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 145-150.  doi: 10.1016/S0764-4442(99)80153-0. [20] A. Miranville, On the conserved phase-field model, J. Math. Anal. Appl., 400 (2013), 143-152.  doi: 10.1016/j.jmaa.2012.11.038. [21] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, Handb. Differ. Equ., 4 (2018), 103-200. doi: 10.1016/S1874-5717(08)00003-0. [22] G. Mola, Global attractors for a three-dimensional conserved phase-field system with memory, Commun. Pure Appl. Anal., 7 (2008), 317-353.  doi: 10.3934/cpaa.2008.7.317. [23] G. Mola, Stability of global and exponential attractors for a three-dimensional conserved phase-field system with memory, Math. Models Methods Appl. Sci., 32 (2009), 2368-2404.  doi: 10.1002/mma.1139. [24] A. Novick-Cohen, On the viscous Cahn-Hilliard equation, in Material Instabilities in Continuum Mechanics, Oxford Univ. Press, New York, 1988. [25] G. Raugel, Singularly perturbed hyperbolic equations revisited, in International Conference on Differential Equations, World Sci. Publishing, River Edge, NJ, 2000. [26] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd Edition, Springer-Verlag, Berlin, Heidelberg, New York, 1997. doi: 10.1007/978-1-4612-0645-3.
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