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- Title
Stability of exponential attractors for singularly perturbed phase-field systems of Oono type.
- Authors
Bonfoh, Ahmed; Alzahrani, Rabab; Miranville, Alain
- Abstract
We consider a phase-field system that takes into account the long-ranged interactions in phase separation based on a theory introduced by Y. Oono et al.$\left\{\begin{array}{l}\tau \phi_t+N(N \phi+g(\phi)-u)+\sigma \phi = 0, \\\varepsilon u_t+\phi_t+N u+\gamma u = 0, \end{array}\right.\;\;\;\;\;\;\;\;\;\;\;\;\;\;(\mathrm{S}_{\varepsilon})$where $ \tau>0 $, $ \sigma, \gamma\ge 0 $, $ \varepsilon\in (0, 1] $ is the heat capacity, $ \phi $ is the order parameter, $ u $ is the absolute temperature, and 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. We consider a class of nonlinear functions $ g\in{\mathcal C}^{3}(\mathbb R) $ that contains the polynomial $ g(\phi) = \sum_{k = 1}^{2p-1}a_k\phi^k, $ $ p\in{\mathbb N}, $ $ p\ge 2, $ $ a_{2p-1}>0 $. We prove a well-posedness result and the existence of the global attractor which is upper semicontinuous at $ \varepsilon = 0 $. Then we construct a family of exponential attractors that is Hölder continuous at $ \varepsilon = 0 $. Our present contribution completes and generalizes some recent results proven by Bonfoh and Suleman in [Comm. Pure Appl. Anal. 2021; 20: 3655-3682] where a conserved model (corresponding to $ \sigma = \gamma = 0 $ in (S$_\varepsilon$)) that takes into account the viscosity effects in the material was considered.
- Subjects
EXPONENTIAL stability; SINGULAR perturbations; NEUMANN boundary conditions; PHASE space; HOLDER spaces; HEAT capacity; NONLINEAR functions
- Publication
Evolution Equations & Control Theory, 2024, Vol 13, Issue 3, p1
- ISSN
2163-2480
- Publication type
Article
- DOI
10.3934/eect.2024008