On the Cahn-Hilliard equation in H-1(R-N)
| dc.contributor.author | Cholewa, Jan W. | |
| dc.contributor.author | Rodríguez Bernal, Aníbal | |
| dc.date.accessioned | 2023-06-20T00:21:02Z | |
| dc.date.available | 2023-06-20T00:21:02Z | |
| dc.date.issued | 2012-12 | |
| dc.description.abstract | In this paper we exhibit the dissipative mechanism of the Cahn-Hilliard equation in H-1 (R-N). We show a weak form of dissipativity by showing that each individual solution is attracted, in some sense, by the set of equilibria. We also indicate that strong dissipativity, that is, asymptotic compactness in H-1 (R-N), cannot be in general expected. Then we consider two types of perturbations: a nonlinear perturbation and a small linear perturbation. In both cases we show that, for the resulting equations, the dissipative mechanism becomes strong enough to obtain the existence of a compact global attractor. | |
| dc.description.department | Depto. de Análisis Matemático y Matemática Aplicada | |
| dc.description.faculty | Fac. de Ciencias Matemáticas | |
| dc.description.refereed | TRUE | |
| dc.description.sponsorship | MEC | |
| dc.description.sponsorship | UCM | |
| dc.description.status | pub | |
| dc.eprint.id | https://eprints.ucm.es/id/eprint/17430 | |
| dc.identifier.doi | 10.1016/j.jde.2012.08.033 | |
| dc.identifier.issn | 0022-0396 | |
| dc.identifier.officialurl | http://www.sciencedirect.com/science/article/pii/S0022039612003476 | |
| dc.identifier.relatedurl | http://www.sciencedirect.com/ | |
| dc.identifier.uri | https://hdl.handle.net/20.500.14352/42439 | |
| dc.issue.number | 12 | |
| dc.journal.title | Journal of Differential Equations | |
| dc.language.iso | eng | |
| dc.page.final | 3726 | |
| dc.page.initial | 3678 | |
| dc.publisher | Elsevier | |
| dc.relation.projectID | GR58/08 Grupo 920894 | |
| dc.relation.projectID | MTM2009-07540 | |
| dc.rights.accessRights | restricted access | |
| dc.subject.cdu | 517.98 | |
| dc.subject.keyword | Cahn-Hilliard equation | |
| dc.subject.keyword | Initial value problems for higher-order | |
| dc.subject.keyword | parabolic equations | |
| dc.subject.keyword | Semilinear parabolic equations | |
| dc.subject.keyword | Asymptotic behavior of solutions | |
| dc.subject.keyword | Attractors | |
| dc.subject.ucm | Análisis funcional y teoría de operadores | |
| dc.title | On the Cahn-Hilliard equation in H-1(R-N) | |
| dc.type | journal article | |
| dc.volume.number | 253 | |
| dcterms.references | H. Amann, Linear and Quasilinear Parabolic Problems, Birkhäuser, Basel, 1995. J.M. Arrieta, A.N. Carvalho, Abstract parabolic problems with critical nonlinearities and applications to Navier–Stokes and heat equations, Trans. Amer. Math. Soc. 352 (2000) 285–310. J.M. Arrieta, A.N. Carvalho, A. Rodriguez-Bernal, Parabolic problems with nonlinear boundary conditions and critical nonlinearities, J. Differential Equations 156 (1999) 376–406. J.M. Arrieta, J.W. Cholewa, T. Dlotko, A. Rodriguez-Bernal, Asymptotic behavior and attractors for reaction–diffusion equations in unbounded domains, Nonlinear Anal. TMA 56 (2004) 515–554. J.M. Arrieta, J.W. Cholewa, T. Dlotko, A. Rodriguez-Bernal, Linear parabolic equations in locally uniform spaces, Math. Models Methods Appl. Sci. 14 (2004) 253–294. A.V. Babin, M.I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992. D. Blömker, S. Maier–Paape, T. Wanner, Spinoidal decomposition for the Cahn–Hilliard–Cook equation, Comm. Math. Phys. 223 (2001) 553–582. D. Blömker, S. Maier–Paape, T. Wanner, Second phase spinoidal decomposition for the Cahn–Hilliard–Cook equation, Trans. Amer. Math. Soc. 360 (2008) 449–489. A. Bonfoh, Finite-dimensional attractor for the viscous Cahn–Hilliard equation in an unbounded domain, Quart. Appl. Math. 65 (2006) 93–104. J. Bricmont, A. Kupiainen, J. Taskinen, Stability of Cahn–Hilliard fronts, Comm. Pure Appl. Math. 52 (1999) 839–871. J.W. Cahn, J.E. Hilliard, Free energy of a nonuniform system, I. Interfacial free energy, J. Chem. Phys. 28 (1958) 258–267. A.N. Carvalho, J.W. Cholewa, Strongly damped wave equations in W1,p 0 (Ω) × Lp(Ω), Discrete Contin. Dyn. Syst. (Suppl.) (2007) 230–239. A.N. Carvalho, T. Dlotko, Dynamics of the viscous Cahn–Hilliard equation, J. Math. Anal. Appl. 344 (2008) 703–725. J.W. Cholewa, T. Dlotko, Global attractor for the Cahn–Hilliard system, Bull. Aust. Math. Soc. 49 (1994) 277–293. J.W. Cholewa, T. Dlotko, Global Attractors in Abstract Parabolic Problems, Cambridge University Press, Cambridge, 2000. J.W. Cholewa, A. Rodriguez-Bernal, Linear and semilinear higher order parabolic equations in RN , Nonlinear Anal. 75 (2012) 194–210. J.W. Cholewa, A. Rodriguez-Bernal, Dissipative mechanism of a semilinear higher order parabolic equation in RN, Nonlinear Anal. 75 (2012) 3510–3530. T. Dlotko, C. Sun, Dynamics of the modified viscous Cahn–Hilliard equation in RN , Topol. Methods Nonlinear Anal. 35 (2010) 277–294. T. Dlotko, M. Kania, C. Sun, Analysis of the viscous Cahn–Hilliard equation in RN , J. Differential Equations 252 (3) (2012) 2771–2791. L. Duan, S. Liu, H. Zhao, A note on the optimal temporal decay estimates of solutions to the Cahn–Hilliard equation, J. Math. Anal. Appl. 372 (2010) 666–678. C.M. Elliott, A.M. Stuart, Viscous Cahn–Hilliard equation II. Analysis, J. Differential Equations 128 (1996) 387–414. K.-J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000. D.J. Eyre, Systems of Cahn–Hilliard equations, University of Minnesota, AHPCRC preprint 92-102, 1992. M. Grasselli, G. Schimperna, S. Zelik, Trajectory and smooth attractors for Cahn–Hilliard equations with inertial term, Nonlinearity 23 (2010) 707–737. J.K. Hale, Asymptotic Behavior of Dissipative Systems, AMS, Providence, RI, 1988. D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, 1981. A. Jiménez–Casas, A. Rodríguez–Bernal, Asymptotic behaviour for a phase field model in higher order Sobolev spaces, Rev. Mat. Complut. 15 (2002) 213–248. T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ration. Mech. Anal. 58 (1975) 181–205. O. Kavian, Introduction a la theorie des points critiques, Math. Appl. (Springer), vol. 13, Springer-Verlag, 1993. H. Komatsu, Fractional powers of operators, Pacific J. Math. 19 (1966) 285–346. O.A. Ladyženskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991. De-Sheng Li, Chen-Kui Zhong, Global attractor for the Cahn–Hilliard system with fast growing nonlinearity, J. Differential Equations 149 (1998) 191–210. S. Liu, F. Wang, H. Zhao, Global existence and asymptotics of solutions of the Cahn–Hilliard equation, J. Differential Equations 238 (2007) 426–469. A. Miranville, Long time behavior of some models of Cahn–Hilliard equations in deformable continua, Nonlinear Anal. RWA 2 (2001) 273–304. A. Novick-Cohen, On the viscous Cahn–Hilliard equation, in: Material Instabilities in Continuum Mechanics, Edinburgh, 1985–1986, Oxford Sci. Publ., Oxford Univ. Press, New York, 1988, pp. 329–342. A. Novick-Cohen, The Cahn–Hilliard equation, in: Handbook of Differential Equations: Evolutionary Equations, vol. 4, Elsevier, 2008, pp. 201–228. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983. A. Rodriguez-Bernal, Perturbation of analytic semigroups in scales of Banach spaces and applications to parabolic equations with low regularity data, SEMA J. 53 (2011) 3–54. A. Rodriguez-Bernal, A. Vidal-López, Semistable extremal ground states for nonlinear evolution equations in unbounded domains, J. Math. Anal. Appl. 338 (2008) 675–694. R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Veb Deutscher Verlag, Berlin, 1978, also: North-Holland, Amsterdam, 1978. S. Zheng, A. Milani, Exponential attractors and inertial manifolds for singular perturbations of the Cahn–Hilliard equations, Nonlinear Anal. TMA 57 (2004) 843–877. S. Zheng, A. Milani, Global attractors for singular perturbations of the Cahn–Hilliard equations, J. Differential Equations 209 (2005) 101–139. | |
| dspace.entity.type | Publication | |
| relation.isAuthorOfPublication | fb7ac82c-5148-4dd1-b893-d8f8612a1b08 | |
| relation.isAuthorOfPublication.latestForDiscovery | fb7ac82c-5148-4dd1-b893-d8f8612a1b08 |
Download
Original bundle
1 - 1 of 1