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

Ahmed Bonfoh; Ibrahim A. Suleman

doi:10.3934/cpaa.2021125

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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

Ahmed Bonfoh ^, and Ibrahim A. Suleman

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.

Keywords: Conserved phase-field systems, singular perturbation, global attractor, exponential attractors, continuity.

Mathematics Subject Classification: Primary: 35B25, 35B41; Secondary: 80A22.

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