Global and exponential attractors for a Caginalp type phase-field problem

• Autorzy: Brice Bangola
• Języki publikacji: EN

Abstrakty

EN

We deal with a generalization of the Caginalp phase-field model associated with Neumann boundary conditions. We prove that the problem is well posed, before studying the long time behavior of solutions. We establish the existence of the global attractor, but also of exponential attractors. Finally, we study the spatial behavior of solutions in a semi-infinite cylinder, assuming that such solutions exist.

Słowa kluczowe

EN

Caginalp phase-field model; Neumann boundary conditions; Well-posedness; Long time behavior of solutions; Global attractor; Exponential attractor; Spatial behavior of solutions; Semi-infinite cylinder

Kategorie tematyczne

• 35A01: Existence problems: global existence, local existence, non-existence
• 35A02: Uniqueness problems: global uniqueness, local uniqueness, non-uniqueness
• 35B45: A priori estimates
• 35B65: Smoothness and regularity of solutions
• 35B40: Asymptotic behavior of solutions
• 35B41: Attractors

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2013
• Tom: 11
• Numer: 9
• Strony: 1651-1676

Daty

wydano

2013-09-01

online

2013-06-28

Twórcy

autor

Brice Bangola

• Université de Poitiers

Bibliografia

• [1] Babin A., Nicolaenko B., Exponential attractors of reaction-diffusion systems in an unbounded domain, J. Dynam. Differential Equations, 1995, 7(4), 567–590 http://dx.doi.org/10.1007/BF02218725
• [2] Bates P.W., Zheng S.M., Inertial manifolds and inertial sets for the phase-field equations, J. Dynam. Differential Equations, 1992, 4(2), 375–398 http://dx.doi.org/10.1007/BF01049391
• [3] Brezis H., Analyse Fonctionnelle, Collect. Math. Appl. Maitrise, Masson, Paris, 1983
• [4] Brochet D., Hilhorst D., Universal attractor and inertial sets for the phase field model, Appl. Math. Lett., 1991, 4(6), 59–62, 1991 http://dx.doi.org/10.1016/0893-9659(91)90076-8
• [5] Brochet D., Hilhorst D., Chen X., Finite-dimensional exponential attractor for the phase field model, Appl. Anal., 1993, 49(3–4), 197–212 http://dx.doi.org/10.1080/00036819108840173
• [6] Brochet D., Hilhorst D., Novick-Cohen A., Finite-dimensional exponential attractor for a model for order-disorder and phase separation, Appl. Math. Lett., 1994, 7(3), 83–87 http://dx.doi.org/10.1016/0893-9659(94)90118-X
• [7] Caginalp G., An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 1986, 92(3), 205–245 http://dx.doi.org/10.1007/BF00254827
• [8] Caginalp G., The role of microscopic anisotropy in the macroscopic behavior of a phase boundary, Ann. Physics, 1986, 172(1), 136–155 http://dx.doi.org/10.1016/0003-4916(86)90022-9
• [9] Cherfils L., Miranville A., Some results on the asymptotic behavior of the Caginalp system with singular potentials, Adv. Math. Sci. Appl., 2007, 17(1), 107–129
• [10] Cherfils L., Miranville A., On the Caginalp system with dynamic boundary conditions and singular potentials, Appl. Math., 2009, 54(2), 89–115 http://dx.doi.org/10.1007/s10492-009-0008-6
• [11] Christov C.I., Jordan P.M., Heat conduction paradox involving second-sound propagation in moving media, Phys. Rev. Lett., 2005, 94(15), #154301 http://dx.doi.org/10.1103/PhysRevLett.94.154301
• [12] Conti M., Gatti S., Miranville A., Asymptotic behavior of the Caginalp phase-field system with coupled dynamic boundary conditions, Discrete Contin. Dyn. Syst. Ser., 2012, S5(3), 485–505
• [13] Eden A., Foias C., Nicolaenko B., Temam R., Exponential Attractors for Dissipative Evolution Equations, RAM Res. Appl. Math., 37, Masson/John Wiley & Sons, Paris/Chichester, 1994
• [14] Fabrie P., Galusinski C., Exponential attractors for a partially dissipative reaction system, Asymptotic Anal., 1996, 12(4), 329–354
• [15] Flavin J.N., Knops R.J., Payne L.E., Decay estimates for the constrained elastic cylinder of variable cross section, Quart. Appl. Math., 1989, 47(2), 325–350
• [16] Gurtin M.E., Chen P.J., On a theory of heat conduction involving two temperatures, Z. Angew. Math. Phys., 1968, 19(4), 614–627 http://dx.doi.org/10.1007/BF01594969
• [17] Landau L.D., Lifshitz E.M., Statistical Physics I, 3rd ed., Butterworth-Heinemann, Oxford, 1980
• [18] Lions J.-L., Magenes E., Problèmes aux Limites non Homogènes et Applications, 2, Travaux et Recherches Mathématiques, 18, Dunod, Paris, 1968
• [19] Miranville A., Exponential attractors for a class of evolution equations by a decomposition method, C. R. Acad. Sci. Paris Sér. I Math., 1999, 328(2), 145–150 http://dx.doi.org/10.1016/S0764-4442(99)80153-0
• [20] Miranville A., Some models of Cahn-Hilliard equations in nonisotropic media, M2AN Math. Model. Numer. Anal., 2000, 34(3), 539–554 http://dx.doi.org/10.1051/m2an:2000155
• [21] Miranville A., Some mathematical models in phase transition, Ravello, 2009 (lecture notes)
• [22] Miranville A., Quintanilla R., A generalization of the Caginalp phase-field system based on the Cattaneo law, Nonlinear Anal., 2009, 71(5–6), 2278–2290 http://dx.doi.org/10.1016/j.na.2009.01.061
• [23] Miranville A., Quintanilla R., Some generalizations of the Caginalp phase-field system, Appl. Anal., 2009, 88(6), 897–894 http://dx.doi.org/10.1080/00036810903042182
• [24] Miranville A., Quintanilla R., A Caginalp phase-field system with a nonlinear coupling, Nonlinear Anal. Real World Appl., 2010, 11(4), 2849–2861 http://dx.doi.org/10.1016/j.nonrwa.2009.10.008
• [25] Miranville A., Zelik S., Attractors for dissipative partial differential equations in bounded and unbounded domains, In: Handbook of Differential Equations: Evolutionary Equations, IV, Handb. Diff. Equ., Elsevier/North-Holland, Amsterdam, 2008, 103–200 http://dx.doi.org/10.1016/S1874-5717(08)00003-0
• [26] Quintanilla R., Spatial stability for the quasi-static problem of thermoelasticity, J. Elasticity, 2004, 76(2), 93–105 http://dx.doi.org/10.1007/s10659-004-3334-7
• [27] Temam R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed., Appl. Math. Sci., 68, Springer, New York, 1997

Identyfikatory

DOI

10.2478/s11533-013-0258-0