Advanced search
Publication Cover
Applicable Analysis
An International Journal
Volume 49, 1993 - Issue 3-4
Submit an article Journal homepage
46
Views
62
CrossRef citations to date
0
Altmetric
Original Articles

Finite dimensional exponential attractor for the phase field model

D. Brochet Laboratoire d'Analyse Numérique, Université Paris Sud, Bâtiment, Orsay, 425 91405, France
,
D. Hilhorst Laboratoire d'Analyse Numérique, Université Paris Sud, Bâtiment, Orsay, 425 91405, France
&
Xinfu chen Department of Mathematics & Statistics, University of Pittsburgh, Pittsburgh, Pennsylvania, 15260, USA
Pages 197-212 | Received 15 Jun 1992, Published online: 02 May 2007

Citations (62)

Keep up to date with the latest research on this topic with citation updates for this article.

Subscribe to citation updates

Read on this site (2)

Manman Yang & Li Ma. (2023) Well-posedness of solutions to a phase-field model for the martensitic phase transformations. Applicable Analysis 102:9, pages 2403-2417.
Read now
Alain Miranville & Ramon Quintanilla. (2009) Some generalizations of the Caginalp phase-field system. Applicable Analysis 88:6, pages 877-894.
Read now

Articles from other publishers (60)

M. BOUGUEZZI, D. HILHORST, Y. MIYAMOTO & J.-F. SCHEID. (2022) Convergence to a self-similar solution for a one-phase Stefan problem arising in corrosion theory. European Journal of Applied Mathematics 34:4, pages 701-737.
Crossref
Manman Yang, Fan Wu & Zixian Zhu. (2022) Global existence of weak solution to a phase‐field model on martensitic phase transformations. Mathematical Methods in the Applied Sciences 46:4, pages 3545-3559.
Crossref
Yangxin Tang & Lin Zheng. (2022) Global existence and large‐time behavior of a parabolic phase‐field model with Neumann boundary conditions. Mathematical Methods in the Applied Sciences 46:1, pages 339-355.
Crossref
Armel Judice Ntsokongo & Christian Tathy. (2022) Long-time behavior of the higher-order anisotropic Caginalp phase-field systems based on the Cattaneo law. Asymptotic Analysis 128:1, pages 1-30.
Crossref
Urbain Cyriaque Mavoungou, Narcisse Batangouna, Franck Davhys Reval Langa, Daniel Moukoko & Macaire Batchi. (2021) Study of the dissipativity, global attractor and exponential attractor for a hyperbolic relaxation of the Caginalp phase-field system with singular nonlinear terms. Asymptotic Analysis 125:1-2, pages 159-186.
Crossref
Jesus Ildefonso Diaz, Danielle Hilhorst & Paris Kyriazopoulos. (2021) parabolic system with strong absorption modeling dry-land vegetation. Electronic Journal of Differential Equations 2021:01-104, pages 08.
Crossref
Perla El Kettani. (2020) Well‐posedness of a stochastic phase‐field problem with multiplicative noises. Mathematical Methods in the Applied Sciences 43:15, pages 8538-8567.
Crossref
H. ABELS & J. KAMPMANN. (2019) On a model for phase separation on biological membranes and its relation to the Ohta–Kawasaki equation. European Journal of Applied Mathematics 31:2, pages 297-338.
Crossref
Aymard Christbert Nimi & Daniel Moukoko. (2020) Global attractor and exponential attractor for a Parabolic system of Cahn-Hilliard with a proliferation term. AIMS Mathematics 5:2, pages 1383-1399.
Crossref
Alain Miranville. (2016) On Higher-Order Anisotropic Conservative Caginalp Phase-Field Systems. Applied Mathematics & Optimization 77:2, pages 297-314.
Crossref
Ján Eliaš, M. Humayun Kabir & Masayasu Mimura. (2017) On the well-posedness of a dispersal model for farmers and hunter–gatherers in the Neolithic transition. Mathematical Models and Methods in Applied Sciences 28:02, pages 195-222.
Crossref
Narcisse Batangouna & Morgan Pierre. (2018) Convergence of exponential attractors for a time splitting approximation of the Caginalp phase-field system. Communications on Pure & Applied Analysis 17:1, pages 1-19.
Crossref
M. Conti, S. Gatti, A. Miranville & R. Quintanilla. (2017) On a Caginalp Phase-Field System with Two Temperatures and Memory. Milan Journal of Mathematics 85:1, pages 1-27.
Crossref
Armel Judice Ntsokongo. (2017) ON HIGHER-ORDER ANISOTROPIC CAGINALP PHASE-FIELD SYSTEMS WITH POLYNOMIAL NONLINEAR TERMS. Journal of Applied Analysis & Computation 7:3, pages 992-1012.
Crossref
Alain Miranville. (2016) Higher-Order Anisotropic Caginalp Phase-Field Systems. Mediterranean Journal of Mathematics 13:6, pages 4519-4535.
Crossref
Alain Miranville & Ramon Quintanilla. (2016) On the Caginalp phase‐field systems with two temperatures and the Maxwell–Cattaneo law. Mathematical Methods in the Applied Sciences 39:15, pages 4385-4397.
Crossref
Charbel Wehbe. (2015) On a Caginalp phase-field system with a logarithmic nonlinearity. Applications of Mathematics 60:4, pages 355-382.
Crossref
Alain Miranville. (2015) A reformulation of the Caginalp phase-field system based on the Maxwell–Cattaneo law. International Journal of Engineering Science 88, pages 128-140.
Crossref
Samira Boussaïd, Danielle Hilhorst & Thanh Nam Nguyen. (2015) Convergence to steady state for the solutions of a nonlocal reaction-diffusion equation. Evolution Equations and Control Theory 4:1, pages 39-59.
Crossref
F. D. Araruna, J. L. Boldrini & B. M. R. Calsavara. (2014) Optimal Control and Controllability of a Phase Field System with One Control Force. Applied Mathematics & Optimization 70:3, pages 539-563.
Crossref
Alain Miranville. (2013) Existence of Strong Solutions for a Fully Hyperbolic Phase-Field Model Based on Type III Heat Conduction with a Logarithmic Nonlinear Term. Mediterranean Journal of Mathematics 11:4, pages 1129-1148.
Crossref
Alain Miranville. (2014) Some mathematical models in phase transition. Discrete & Continuous Dynamical Systems - S 7:2, pages 271-306.
Crossref
Bin Wu, Qun Chen & Zewen Wang. (2013) Uniqueness and stability of an inverse problem for a phase field model using data from one component. Computers & Mathematics with Applications 66:10, pages 2126-2138.
Crossref
Harald Garcke. (2013) Curvature Driven Interface Evolution. Jahresbericht der Deutschen Mathematiker-Vereinigung 115:2, pages 63-100.
Crossref
Brice Bangola. (2013) Global and exponential attractors for a Caginalp type phase-field problem. Open Mathematics 11:9.
Crossref
Pedro Marín-Rubio & Gabriela Planas. (2012) Global attractor and omega-limit sets structure for a phase-field model of thermal alloys. Nonlinear Analysis: Real World Applications 13:4, pages 1676-1691.
Crossref
Alain Miranville. (2012) On a phase-field model with a logarithmic nonlinearity. Applications of Mathematics 57:3, pages 215-229.
Crossref
Alain Miranville & Ramon Quintanilla. (2012) On a phase-field system based on the Cattaneo law. Nonlinear Analysis: Theory, Methods & Applications 75:4, pages 2552-2565.
Crossref
. (2012) ATTRACTORS FOR A CAGINALP PHASE-FIELD MODEL TYPE ON THE WHOLE SPACE R<sup>3</sup>. Journal of Applied Analysis & Computation 2:3, pages 251-272.
Crossref
Alain Miranville & Ramon Quintanilla. (2010) A Phase-Field Model Based on a Three-Phase-Lag Heat Conduction. Applied Mathematics & Optimization 63:1, pages 133-150.
Crossref
Ahmed Bonfoh. (2010) The singular limit dynamics of the phase-field equations. Annali di Matematica Pura ed Applicata 190:1, pages 105-144.
Crossref
F. BALIBREA, T. CARABALLO, P. E. KLOEDEN & J. VALERO. (2012) RECENT DEVELOPMENTS IN DYNAMICAL SYSTEMS: THREE PERSPECTIVES. International Journal of Bifurcation and Chaos 20:09, pages 2591-2636.
Crossref
Alain Miranville & Ramon Quintanilla. (2010) A Caginalp phase-field system with a nonlinear coupling. Nonlinear Analysis: Real World Applications 11:4, pages 2849-2861.
Crossref
Masaki Kurokiba, Naoto Tanaka & Atusi Tani. (2010) Maximal attractor and inertial set for Eguchi–Oki–Matsumura equation. Journal of Mathematical Analysis and Applications 365:2, pages 638-645.
Crossref
Cecilia Cavaterra, Ciprian G. Gal, Maurizio Grasselli & Alain Miranville. (2010) Phase-field systems with nonlinear coupling and dynamic boundary conditions. Nonlinear Analysis: Theory, Methods & Applications 72:5, pages 2375-2399.
Crossref
Danielle Hilhorst & Piotr Rybka. (2009) Stabilization of Solutions to a FitzHugh-Nagumo Type System. Journal of Statistical Physics 138:1-3, pages 291-304.
Crossref
Alain Miranville & Ramon Quintanilla. (2009) A generalization of the Caginalp phase-field system based on the Cattaneo law. Nonlinear Analysis: Theory, Methods & Applications 71:5-6, pages 2278-2290.
Crossref
José Luiz Boldrini, Bianca Morelli Calsavara Caretta & Enrique Fernández-Cara. (2009) Analysis of a two-phase field model for the solidification of an alloy. Journal of Mathematical Analysis and Applications 357:1, pages 25-44.
Crossref
Takashi Suzuki & Souhei Tasaki. (2009) Stationary Fix–Caginalp equation with non-local term. Nonlinear Analysis: Theory, Methods & Applications 71:3-4, pages 1329-1349.
Crossref
Ciprian G. Gal & Alain Miranville. (2009) Uniform global attractors for non-isothermal viscous and non-viscous Cahn–Hilliard equations with dynamic boundary conditions. Nonlinear Analysis: Real World Applications 10:3, pages 1738-1766.
Crossref
Pedro Marín-Rubio, Gabriela Planas & José Real. (2009) Asymptotic behaviour of a phase-field model with three coupled equations without uniqueness. Journal of Differential Equations 246:12, pages 4632-4652.
Crossref
Jie Jiang. (2008) Convergence to equilibrium for a fully hyperbolic phase‐field model with Cattaneo heat flux law. Mathematical Methods in the Applied Sciences 32:9, pages 1156-1182.
Crossref
L. Cherfils & A. Miranville. (2009) On the Caginalp system with dynamic boundary conditions and singular potentials. Applications of Mathematics 54:2, pages 89-115.
Crossref
Maurizio Grasselli, Hao Wu & Songmu Zheng. (2008) Asymptotic behavior of a nonisothermal Ginzburg-Landau model. Quarterly of Applied Mathematics 66:4, pages 743-770.
Crossref
Francisco Morillas & José Valero. (2008) Asymptotic Compactness and Attractors for Phase-Field Equations in ℝ3. Set-Valued Analysis 16:7-8, pages 861-897.
Crossref
Laurence Cherfils, Stefania Gatti & Alain Miranville. (2008) Existence of global solutions to the Caginalp phase-field system with dynamic boundary conditions and singular potentials. Journal of Mathematical Analysis and Applications 343:1, pages 557-566.
Crossref
Jie Jiang. (2008) Convergence to equilibrium for a parabolic–hyperbolic phase field model with Cattaneo heat flux law. Journal of Mathematical Analysis and Applications 341:1, pages 149-169.
Crossref
Hao Wu, Maurizio Grasselli & Songmu Zheng. (2007) Convergence to equilibrium for a parabolic–hyperbolic phase-field system with dynamical boundary condition. Journal of Mathematical Analysis and Applications 329:2, pages 948-976.
Crossref
HAO WU, MAURIZIO GRASSELLI & SONGMU ZHENG. (2011) CONVERGENCE TO EQUILIBRIUM FOR A PARABOLIC–HYPERBOLIC PHASE-FIELD SYSTEM WITH NEUMANN BOUNDARY CONDITIONS. Mathematical Models and Methods in Applied Sciences 17:01, pages 125-153.
Crossref
Erik Burman & Jacques Rappaz. (2003) Existence of solutions to an anisotropic phase‐field model. Mathematical Methods in the Applied Sciences 26:13, pages 1137-1160.
Crossref
Sergiu Aizicovici & Hana Petzeltová. (2003) Asymptotic Behavior of Solutions of a Conserved Phase-Field System with Memory. Journal of Integral Equations and Applications 15:3.
Crossref
Amy Novick-Cohen. (2002) A Phase Field System with Memory: Global Existence. Journal of Integral Equations and Applications 14:1.
Crossref
Pavel Krejčí, Jürgen Sprekels & Songmu Zheng. (2001) Asymptotic Behaviour for a Phase-Field System with Hysteresis. Journal of Differential Equations 175:1, pages 88-107.
Crossref
Sergiu Aizicovici, Eduard Feireisl & Françoise Issard‐Roch. (2001) Long‐time convergence of solutions to a phase‐field system. Mathematical Methods in the Applied Sciences 24:5, pages 277-287.
Crossref
J. Rappaz & J. F. Scheid. (2000) Existence of solutions to a phase-field model for the isothermal solidification process of a binary alloy. Mathematical Methods in the Applied Sciences 23:6, pages 491-513.
Crossref
A. Jiménez-Casas & A. Rodríguez-Bernal. (2011) Linear stability analysis and metastable solutions for a phase-field model. Proceedings of the Royal Society of Edinburgh: Section A Mathematics 129:3, pages 571-600.
Crossref
Ph. Laurençot. (1997) Weak solutions to a phase-field model with non-constant thermal conductivity. Quarterly of Applied Mathematics 55:4, pages 739-760.
Crossref
Sergiu Aizicovici & Viorel Barbu. (1996) Existence and asymptotic results for a system of integro-partial differential equations. Nonlinear Differential Equations and Applications NoDEA 3:1, pages 1-18.
Crossref
Ph. Laurençot. (2011) Long-time behaviour for a model of phase-field type. Proceedings of the Royal Society of Edinburgh: Section A Mathematics 126:1, pages 167-185.
Crossref
D. Brochet, D. Hilhorst & A. Novick-Cohen. (1994) Finite-dimensional exponential attractor for a model for order-disorder and phase separation. Applied Mathematics Letters 7:3, pages 83-87.
Crossref

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Order Reprints Request Corporate Permissions

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

Request Academic Permissions

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.

Related research

People also read lists articles that other readers of this article have read.

Recommended articles lists articles that we recommend and is powered by our AI driven recommendation engine.

Cited by lists all citing articles based on Crossref citations.
Articles with the Crossref icon will open in a new tab.