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MEMORY RELAXATION OF THE ONE-DIMENSIONAL CAHN-HILLIARD EQUATION

2006; World Scientific; Linguagem: Inglês

10.1142/9789812774293_0006

1793-0901

Stefania Gatti, Maurizio Grasselli, Alain Miranville, Vittorino Pata,

Numerical methods in inverse problems

Series on Advances in Mathematics for Applied SciencesDissipative Phase Transitions, pp. 101-114 (2006) No AccessMEMORY RELAXATION OF THE ONE-DIMENSIONAL CAHN-HILLIARD EQUATIONStefania Gatti, Maurizio Grasselli, Alain Miranville, and Vittorino PataStefania GattiDipartimento di Matematica, Università di Ferrara, Via Machiavelli 35, I-44100 Ferrara, Italy, Maurizio GrasselliDipartimento di Matematica "F.Brioschi", Politecnico di Milano, Via Bonardi 9, I-20133 Milano, Italy, Alain MiranvilleLaboratoire de Mathématiques et Applications, Université de Poitiers, Boulevard Marie et Pierre Curie - Téléport 2, F-86962 Chasseneuil Futuroscope Cedex, France, and Vittorino PataDipartimento di Matematica "F.Brioschi", Politecnico di Milano, Via Bonardi 9, I-20133 Milano, Italyhttps://doi.org/10.1142/9789812774293_0006Cited by:9 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail Abstract: We consider the memory relaxation of the one-dimensional Cahn-Hilliard equation, endowed with the no-flux boundary conditions. The resulting integrodifferential equation is characterized by a memory kernel, which is the rescaling of a given positive decreasing function. The Cahn-Hilliard equation is then viewed as the formal limit of the relaxed equation, when the scaling parameter (or relaxation time) ε tends to zero. In particular, if the memory kernel is the decreasing exponential, then the relaxed equation is equivalent to the standard hyperbolic relaxation. The main result of this paper is the existence of a family of robust exponential attractors for the one-parameter dissipative dynamical system generated by the relaxed equation. Such a family is stable with respect to the singular limit ε → 0. FiguresReferencesRelatedDetailsCited By 9Global attractors for the 2D hyperbolic Cahn–Hilliard equationsAzer Khanmamedov and Sema Yayla22 January 2018 | Zeitschrift für angewandte Mathematik und Physik, Vol. 69, No. 1Global well-posedness and attractors for the hyperbolic Cahn–Hilliard–Oono equation in the whole spaceAnton Savostianov and Sergey Zelik5 May 2016 | Mathematical Models and Methods in Applied Sciences, Vol. 26, No. 07Finite dimensionality of the attractor for the hyperbolic Cahn-Hilliard-Oono equation in R3Anton Savostianov and Sergey Zelik10 August 2015 | Mathematical Methods in the Applied Sciences, Vol. 39, No. 5Singular limit of viscous Cahn-Hilliard equations with memory and dynamic boundary conditionsCiprian G. Gal and Maurizio Grasselli1 Jan 2013 | Discrete & Continuous Dynamical Systems - B, Vol. 18, No. 6The Cahn-Hilliard Equation with Logarithmic PotentialsLaurence Cherfils, Alain Miranville and Sergey Zelik28 August 2011 | Milan Journal of Mathematics, Vol. 79, No. 2Singularly perturbed 1D Cahn–Hilliard equation revisitedAhmed Bonfoh, Maurizio Grasselli and Alain Miranville30 April 2010 | Nonlinear Differential Equations and Applications NoDEA, Vol. 17, No. 6Trajectory and smooth attractors for Cahn–Hilliard equations with inertial termMaurizio Grasselli, Giulio Schimperna and Sergey Zelik11 February 2010 | Nonlinearity, Vol. 23, No. 3On the 3D Cahn–Hilliard equation with inertial termMaurizio Grasselli, Giulio Schimperna, Antonio Segatti and Sergey Zelik4 April 2009 | Journal of Evolution Equations, Vol. 9, No. 2Long time behavior of a singular perturbation of the viscous Cahn–Hilliard–Gurtin equationAhmed Bonfoh, Maurizio Grasselli and Alain Miranville1 January 2008 | Mathematical Methods in the Applied Sciences, Vol. 31, No. 6 Dissipative Phase TransitionsMetrics History PDF download