# zbMATH — the first resource for mathematics

A Cahn-Hilliard model in a domain with non-permeable walls. (English) Zbl 1228.35051
Let $$\Omega$$ be a bounded domain of $$\mathbb R^3$$ with smooth boundary $$\Gamma$$ and consider the following Cahn-Hilliard equation
$\partial_t\rho - \Delta\mu = 0 , \quad \mu = -\Delta\rho + f(\rho) \quad\text{in }(0,\infty)\times \Omega$ with dynamic boundary conditions
$w\partial_t \rho - \delta \Delta_\Gamma \mu + \partial_n \mu = \partial_n\rho - \sigma\Delta_\Gamma \rho + g(\rho) - w\mu = 0 \quad \text{on } (0,\infty)\times\Gamma.$ Here $$\delta$$ and $$\sigma$$ are nonnegative real numbers, $$w$$ is a positive bounded function such that $$1/w$$ is bounded, $$\Delta_\Gamma$$ denotes the Laplace-Beltrami operator on $$\Gamma$$, $$f$$ is the derivative of a (possibly non-smooth) double-well potential and $$g$$ is a smooth function. The existence and uniqueness of a weak solution $$\rho$$ are proved for a rather general class of data $$f$$ and $$g$$. Regularity for positive times of the solution is investigated according to the growth of $$f$$ and $$g$$. Existence of a bounded absorbing set and a compact attractor is shown as well. Finally, if $$f$$ and $$g$$ are also assumed to be real analytic functions, convergence of the solution to a steady state is proved with the help of the now classical Łojasiewicz-Simon inequality.

##### MSC:
 35B40 Asymptotic behavior of solutions to PDEs 35K52 Initial-boundary value problems for higher-order parabolic systems 35B41 Attractors 80A22 Stefan problems, phase changes, etc.
Full Text:
##### References:
 [1] Cahn, J.W., On spinodal decomposition, Acta metall., 9, 795-801, (1961) [2] Cahn, J.W.; Hilliard, J.E., Free energy of a nonuniform system. I. interfacial free energy, J. chem. phys., 28, 258-267, (1958) [3] Novick-Cohen, A., The cahn – hilliard equation, (), 201-228 · Zbl 1185.35001 [4] Fischer, H.P.; Maass, P.; Dieterich, W., Novel surface modes in spinodal decomposition, Phys. rev. lett., 79, 893-896, (1997) [5] Fischer, H.P.; Maass, P.; Dieterich, W., Diverging time and length scales of spinodal decomposition modes in thin flows, Europhys. lett., 42, 49-54, (1998) [6] Fischer, H.P.; Reinhard, J.; Dieterich, W.; Gouyet, J.-F.; Maass, P.; Majhofer, A.; Reinel, D., Time-dependent density functional theory and the kinetics of lattice gas systems in contact with a wall, J. chem. phys., 108, 3028-3037, (1998) [7] Chill, R.; Fašangová, E.; Prüss, J., Convergence to steady states of solutions of the cahn – hilliard equation with dynamic boundary conditions, Math. nachr., 279, 1448-1462, (2006) · Zbl 1107.35058 [8] Gilardi, G.; Miranville, A.; Schimperna, G., On the cahn – hilliard equation with irregular potentials and dynamic boundary conditions, Commun. pure appl. anal., 8, 881-912, (2009) · Zbl 1172.35417 [9] Gilardi, G.; Miranville, A.; Schimperna, G., Long-time behavior of the cahn – hilliard equation with irregular potentials and dynamic boundary conditions, Chin. ann. math. ser. B, 31, 679-712, (2010) · Zbl 1223.35067 [10] Miranville, A.; Zelik, S., Exponential attractors for the cahn – hilliard equation with dynamic boundary conditions, Math. methods appl. sci., 28, 709-735, (2005) · Zbl 1068.35020 [11] Miranville, A.; Zelik, S., The cahn – hilliard equation with singular potentials and dynamic boundary conditions, Discrete contin. dyn. syst., 28, 275-310, (2010) · Zbl 1203.35046 [12] Prüss, J.; Racke, R.; Zheng, S., Maximal regularity and asymptotic behavior of solutions for the cahn – hilliard equation with dynamic boundary conditions, Ann. mat. pura appl. (4), 185, 627-648, (2006) · Zbl 1232.35081 [13] Racke, R.; Zheng, S., The cahn – hilliard equation with dynamic boundary conditions, Adv. differential equations, 8, 83-110, (2003) · Zbl 1035.35050 [14] Cherfils, L.; Gatti, S.; Miranville, A., Corrigendum to “existence of global solutions to the Caginalp phase-field system with dynamic boundary conditions and singular potentials” [J. math. anal. appl. 343 (2008) 557-566], J. math. anal. appl., 348, 1029-1030, (2008) · Zbl 1160.35433 [15] Cherfils, L.; Miranville, A., On the Caginalp system with dynamic boundary conditions and singular potentials, Appl. math., 54, 89-115, (2009) · Zbl 1212.35012 [16] Gal, C.G.; Grasselli, M., The nonisothermal allen – cahn equation with dynamic boundary conditions, Discrete contin. dyn. syst. A, 22, 1009-1040, (2008) · Zbl 1160.35353 [17] Gal, C.G.; Grasselli, M., On the asymptotic behavior of the Caginalp system with dynamic boundary conditions, Commun. pure appl. anal., 8, 689-710, (2009) · Zbl 1171.35337 [18] Gal, C.G.; Grasselli, M.; Miranville, A., Nonisothermal allen – cahn equations with coupled dynamic boundary conditions, (), 117-139 · Zbl 1178.35076 [19] Gatti, S.; Miranville, A., Asymptotic behavior of a phase-field system with dynamic boundary conditions, (), 149-170 · Zbl 1123.35310 [20] Grasselli, M.; Miranville, A.; Schimperna, G., The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials, Discrete contin. dyn. syst., 28, 67-98, (2010) · Zbl 1194.35074 [21] Gal, C.G., A cahn – hilliard model in bounded domains with permeable walls, Math. methods appl. sci., 29, 2009-2036, (2006) · Zbl 1109.35057 [22] Gal, C.G., Exponential attractors for a cahn – hilliard model in bounded domains with permeable walls, Electron. J. differential equations, 143, 23 pp, (2006), (electronic) · Zbl 1113.35031 [23] Gal, C.G., Robust exponential attractors for a conserved cahn – hillard model with singularly perturbed boundary conditions, Commun. pure appl. anal., 7, 819-836, (2008) · Zbl 1165.35327 [24] Gal, C.G.; Wu, H., Asymptotic behavior of a cahn – hilliard equation with Wentzell boundary conditions and mass conservation, Discrete contin. dyn. syst., 22, 1041-1063, (2008) · Zbl 1158.35052 [25] Schimperna, G., Abstract approach to evolution equations of phase field type and applications, J. differential equations, 164, 395-430, (2000) · Zbl 0978.35075 [26] Fasano, A.; Primicerio, M.; Rubinstein, L., A model problem for heat conduction with a free boundary in a concentrated capacity, J. inst. math. appl., 26, 327-347, (1980) · Zbl 0456.35093 [27] Magenes, E., Some new results on a Stefan problem in a concentrated capacity, Rend. mat. accad. lincei, IX, 3, 23-34, (1992) · Zbl 0767.35110 [28] Savaré, G.; Visintin, A., Variational convergence of nonlinear diffusion equations: applications to concentrated capacity problems with change of phase, Atti accad. naz. lincei cl. sci. fis. mat. natur. rend. lincei (9) mat. appl., 8, 49-89, (1997) · Zbl 0888.35139 [29] Schimperna, G., Weak solution to a phase-field transmission problem in a concentrated capacity, Math. methods appl. sci., 22, 1235-1254, (1999) · Zbl 0933.35198 [30] Miranville, A.; Zelik, S., Robust exponential attractors for cahn – hilliard type equations with singular potentials, Math. methods appl. sci., 27, 545-582, (2004) · Zbl 1050.35113 [31] Brézis, H., Intégrales convexes dans LES espaces de Sobolev, Israel J. math., 13, 9-23, (1972), (in French) [32] Grun-Rehomme, M., Caractérisation du sous-différentiel d’intégrandes convexes dans LES espaces de Sobolev, J. math. pures appl. (9), 56, 149-156, (1977), (in French) · Zbl 0314.35001 [33] Colli, P.; Gilardi, G.; Grasselli, M.; Schimperna, G., The conserved phase-field system with memory, Adv. math. sci. appl., 11, 265-291, (2001) · Zbl 0982.80006 [34] Rocca, E.; Schimperna, G., Universal attractor for some singular phase transition systems, Physica D, 192, 279-307, (2004) · Zbl 1062.82015 [35] Pata, V.; Zelik, S., A result on the existence of global attractors for semigroups of closed operators, Commun. pure appl. anal., 6, 481-486, (2007) · Zbl 1152.47046 [36] Barbu, V., Nonlinear semigroups and differential equations in Banach spaces, (1976), Noordhoff Leyden [37] Temam, R., Infinite-dimensional dynamical systems in mechanics and physics, (1988), Springer-Verlag New York · Zbl 0662.35001 [38] Geymonat, G., Trace theorems for Sobolev spaces on Lipschitz domains. necessary conditions, Ann. math. blaise Pascal, 14, 187-197, (2007) · Zbl 1161.46019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.