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

Abstract

<p style='text-indent:20px;'>We consider the conserved phase-field system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE111"> \begin{document}$\left\{ \begin{array}{l}\tau {\phi _t} + N(\delta {\phi _t} + N\phi + g(\phi ) - u) = 0,\\\epsilon{u_t} + {\phi _t} + Nu = 0,\end{array} \right.\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {{{\rm{S}}_\varepsilon }} \right)$\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \tau&gt;0 $\end{document}</tex-math></inline-formula> is a relaxation time, <inline-formula><tex-math id="M2">\begin{document}$ \delta&gt;0 $\end{document}</tex-math></inline-formula> is the viscosity parameter, <inline-formula><tex-math id="M3">\begin{document}$ \epsilon\in (0,1] $\end{document}</tex-math></inline-formula> is the heat capacity, <inline-formula><tex-math id="M4">\begin{document}$ \phi $\end{document}</tex-math></inline-formula> is the order parameter, <inline-formula><tex-math id="M5">\begin{document}$ u $\end{document}</tex-math></inline-formula> is the absolute temperature, the Laplace operator <inline-formula><tex-math id="M6">\begin{document}$ N = -\Delta:{\mathscr D}(N)\to \dot L^2(\Omega) $\end{document}</tex-math></inline-formula> is subject to either Neumann boundary conditions (in which case <inline-formula><tex-math id="M7">\begin{document}$ \Omega\subset{\mathbb R}^d $\end{document}</tex-math></inline-formula> is a bounded domain with smooth boundary) or periodic boundary conditions (in which case <inline-formula><tex-math id="M8">\begin{document}$ \Omega = \Pi_{i = 1}^d(0,L_i), $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M9">\begin{document}$ L_i&gt;0 $\end{document}</tex-math></inline-formula>), <inline-formula><tex-math id="M10">\begin{document}$ d = 1,2 $\end{document}</tex-math></inline-formula> or 3, and <inline-formula><tex-math id="M11">\begin{document}$ G(\phi) = \int_0^\phi g(\sigma)d\sigma $\end{document}</tex-math></inline-formula> is a double-well potential. Let <inline-formula><tex-math id="M12">\begin{document}$ j = 1 $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M13">\begin{document}$ d = 1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M14">\begin{document}$ j = 2 $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M15">\begin{document}$ d = 2 $\end{document}</tex-math></inline-formula> or 3. We assume that <inline-formula><tex-math id="M16">\begin{document}$ g\in{\mathcal C}^{j+1}(\mathbb R) $\end{document}</tex-math></inline-formula> and satisfies the conditions <inline-formula><tex-math id="M17">\begin{document}$ g'(\phi)\geq -{\mathscr C}_1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M18">\begin{document}$ G(\phi)\ge -{\mathscr C}_2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M19">\begin{document}$ (\phi-m(\phi))g(\phi)-{\mathscr C}_3(m(\phi))G(s)\ge -{\mathscr C}_4(m(\phi)) $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M20">\begin{document}$ {\mathscr C}_5(\varrho)\le {\mathscr C}_l(m(\phi))\le {\mathscr C}_6(\varrho) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M21">\begin{document}$ l = 3,4 $\end{document}</tex-math></inline-formula>, whenever <inline-formula><tex-math id="M22">\begin{document}$ |m(\phi)|\le \varrho $\end{document}</tex-math></inline-formula>), where <inline-formula><tex-math id="M23">\begin{document}$ \varrho,{\mathscr C}_1, {\mathscr C}_2,{\mathscr C}_4\ge 0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M24">\begin{document}$ {\mathscr C}_3, {\mathscr C}_5,{\mathscr C}_6&gt;0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M25">\begin{document}$ m(\phi) = \frac{1}{|\Omega|}\int_\Omega\phi(x)dx $\end{document}</tex-math></inline-formula>. For instance, <inline-formula><tex-math id="M26">\begin{document}$ g(\phi) = \sum_{k = 1}^{2p-1}a_k\phi^k, $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M27">\begin{document}$ p\in{\mathbb N}, $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M28">\begin{document}$ p\ge 2, $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M29">\begin{document}$ a_{2p-1}&gt;0, $\end{document}</tex-math></inline-formula> satisfies all the above-mentioned conditions. We then prove a well-posedness result, the existence of the global attractor and a family of exponential attractors in the phase space <inline-formula><tex-math id="M30">\begin{document}$ {\mathcal V}_j = {\mathscr D}(N^{j/2})\times{\mathscr D}(N^{j/2}) $\end{document}</tex-math></inline-formula> equipped with the norm <inline-formula><tex-math id="M31">\begin{document}$ \|(\psi,\varphi)\|_{{\mathcal V}_{j}} = (\|N^{j/2}\psi\|^2+m(\psi)^2+\|N^{j/2}\varphi\|^2+m(\varphi)^2)^{1/2} $\end{document}</tex-math></inline-formula>. Moreover, we demonstrate that the global attractor is upper semicontinuous at <inline-formula><tex-math id="M32">\begin{document}$ \epsilon = 0 $\end{document}</tex-math></inline-formula> in the metric induced by the norm <inline-formula><tex-math id="M33">\begin{document}$ \|.\|_{{\mathcal V}_{j+1}} $\end{document}</tex-math></inline-formula>. In addition, the exponential attractors are proven to be Hölder continuous at <inline-formula><tex-math id="M34">\begin{document}$ \epsilon = 0 $\end{document}</tex-math></inline-formula> in the metric induced by the norm <inline-formula><tex-math id="M35">\begin{document}$ \|.\|_{{\mathcal V}_{j}} $\end{document}</tex-math></inline-formula>. Our results improve a recent work by Bonfoh and Enyi [Comm. Pure Appl. Anal. 2016; 35:1077-1105] where the following additional growth condition <inline-formula><tex-math id="M36">\begin{document}$ |g''(\phi)|\leq {\mathscr C}_7\left(|\phi|^{p}+1\right), $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M37">\begin{document}$ {\mathscr C}_7&gt;0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M38">\begin{document}$ p&gt;0 $\end{document}</tex-math></inline-formula> is arbitrary when <inline-formula><tex-math id="M39">\begin{document}$ d = 1, 2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M40">\begin{document}$ p\in [0,3] $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M41">\begin{document}$ d = 3 $\end{document}</tex-math></inline-formula>, was required, preventing <inline-formula><tex-math id="M42">\begin{document}$ g $\end{document}</tex-math></inline-formula> to be a polynomial of any arbitrary odd degree with a strictly positive leading coefficient in three space dimension.</p>