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On a coupled bulk-surface Allen-Cahn system with an affine linear transmission condition and its approximation by a Robin boundary condition. (English) Zbl 1418.35245

Summary: We study a coupled bulk-surface Allen-Cahn system with an affine linear transmission condition, that is, the trace values of the bulk variable and the values of the surface variable are connected via an affine relation, and this serves to generalize the usual dynamic boundary conditions. We tackle the problem of well-posedness via a penalization method using Robin boundary conditions. In particular, for the relaxation problem, the strong well-posedness and long-time behaviour of solutions can be shown for more general and possibly nonlinear relations. New difficulties arise since the surface variable is no longer the trace of the bulk variable, and uniform estimates in the relaxation parameter are scarce. Nevertheless, weak convergence to the original problem with affine linear relations can be shown. Using the approach of the first two authors [Math. Methods Appl. Sci. 38, No. 17, 3950–3967 (2015; Zbl 1334.35165)], we show strong existence to the original problem with affine linear relations, and derive an error estimate between solutions to the relaxed and original problems.

MSC:

35K61 Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35D35 Strong solutions to PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35K86 Unilateral problems for nonlinear parabolic equations and variational inequalities with nonlinear parabolic operators

Citations:

Zbl 1334.35165
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References:

[1] Abels, H.; Garcke, H.; Grün, G., Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flow with different densities, Math. Models Methods Appl. Sci., 22, 3 (2012), 1150013, 40 pp · Zbl 1242.76342
[2] Alikakos, N. D.; Bates, P. W.; Chen, X., The convergence of solutions of the Cahn-Hilliard equation to the solution of Hele-Shaw model, Arch. Ration. Mech. Anal., 128, 165-205 (1994) · Zbl 0828.35105
[3] Allen, S.; Cahn, J., A microscopic theory for the antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall., 27, 6, 1085-1095 (1979)
[4] Babuška, I., The finite element method with penalty, Math. Comp., 27, 122, 221-228 (1973) · Zbl 0299.65057
[5] Barbu, V., Nonlinear Differential Equations of Monotone Types in Banach Spaces (2010), Springer-Verlag, New York · Zbl 1197.35002
[6] Barrett, J. W.; Elliott, C. M., Finite element approximation of the Dirichlet problem using the boundary penalty method, Numer. Math., 49, 343-366 (1986) · Zbl 0614.65116
[7] Brézis, H., Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans le Espaces de Hilbert (1973), North-Holland, Amsterdam · Zbl 0252.47055
[8] Brezis, H.; Crandall, M. G.; Pazy, A., Perturbations of nonlinear maximal monotone sets in Banach spaces, Comm. Pure Appl. Math., 23, 123-144 (1970) · Zbl 0182.47501
[9] Brezzi, F.; Gilardi, G., Partial differential equations, (Kardestuncer, H.; Norrie, D. H., Finite Element Handbook (1987), McGraw-Hill Book Company: McGraw-Hill Book Company New York), 1:77-1:121
[10] Caginalp, G., An analysis of a phase field model of a free boundary, Arch. Ration. Mech. Anal., 92, 205-245 (1986) · Zbl 0608.35080
[11] Cahn, J. W.; Hilliard, J. E., Free energy of a nonuniform system I. interfacial free energy, J. Chem. Phys., 28, 258-267 (1958) · Zbl 1431.35066
[12] Calatroni, L.; Colli, P., Global solution to the Allen-Cahn equation with singular potentials and dynamic boundary conditions, Nonlinear Anal., 79, 12-27 (2013) · Zbl 1270.35275
[13] Chen, X., Global asymptotic limit of solutions of the Cahn-Hilliard equation, J. Differential Geom., 44, 262-311 (1996) · Zbl 0874.35045
[14] Colli, P.; Fukao, T., The Allen-Cahn equation with dynamic boundary conditions and mass constraints, Math. Models Appl. Sci., 38, 17, 3950-3967 (2015) · Zbl 1334.35165
[15] Colli, P.; Gilardi, G.; Nakayashiki, R.; Shirakawa, K., A class of quasi-linear Allen-Cahn type equations with dynamic boundary conditions, Nonlinear Anal., 158, 32-59 (2017) · Zbl 1372.35162
[16] Colli, P.; Visintin, A., On a class of doubly nonlinear evolution equations, Comm. Partial Differential Equations, 15, 5, 737-756 (1990) · Zbl 0707.34053
[17] Dziuk, G.; Elliott, C., Finite element methods for surface PDEs, Acta Numer., 22, 289-396 (2013) · Zbl 1296.65156
[18] Escher, J.; Simonett, G., A center manifold analysis for the Mullins-Sekerka model, J. Differential Equations, 143, 267-292 (1998) · Zbl 0896.35142
[19] Fischer, H. P.; Maass, P.; Dieterich, W., Novel surface modes in spinodal decomposition, Phys. Rev. Lett., 79, 5, 893-896 (1997)
[20] Gal, C. G.; Grasselli, M., The non-isothermal Allen-Cahn equation with dynamic boundary conditions, Discrete Contin. Dyn. Syst., 22, 1009-1040 (2008) · Zbl 1160.35353
[21] 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
[22] Garcke, H.; Kampmann, J.; Rätz, A.; Röger, M., A coupled surface-Cahn-Hilliard bulk-diffusion system modeling lipid raft formation in cell membranes, Math. Methods Models Appl. Sci., 26, 6, 1149-1189 (2016) · Zbl 1338.35222
[23] Garcke, H.; Lam, K. F., Analysis of a Cahn-Hilliard system with non-zero Dirichlet conditions modeling tumor growth with chemotaxis, Discrete Contin. Dyn. Sys., 37, 8, 4277-4308 (2017) · Zbl 1360.35042
[24] Gilardi, G.; Miranville, A.; Schimperna, G., On the Cahn-Hilliard equation with irregular potentials and dynamic boundary equations, Commun. Pure Appl. Anal., 8, 881-912 (2009) · Zbl 1172.35417
[25] Kenzler, R.; Eurich, F.; Maass, P.; Rinn, B.; Schropp, J.; Bohl, E.; Dieterich, W., Phase separation in confined geometries: Solving the Cahn-Hilliard equation with generic boundary conditions, Comput. Phys. Comm., 133, 139-157 (2001) · Zbl 0985.65114
[26] Liu, I.-S., Method of Lagrange multipliers for exploitation of the entropy principle, Arch. Ration. Mech. Anal., 46, 2, 131-148 (1972) · Zbl 0252.76003
[27] Liu, I.-S., Continuum mechanics, (Advanced Texts in Physics (2002), Springer-Verlag: Springer-Verlag Berlin) · Zbl 1058.74004
[28] Liu, C.; Wu, H., An energetic variational approach for the Cahn-Hilliard equation with dynamic boundary conditions: Derivation and analysis, Arch. Ration. Mech. Anal. (2019)
[29] Miranville, A.; Zelik, S., The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions, Discrete Contin. Dyn. Syst., 28, 1, 275-310 (2010) · Zbl 1203.35046
[30] Mullins, W. W.; Sekerka, R. F., Morphological stability of a particle growing by diffusion or heat flow, J. Appl. Phys., 34, 323-329 (1963)
[31] D. O’Connor, B. Stinner, The Cahn-Hilliard equation on an evolving surface, Preprint arXiv:1607.05627; D. O’Connor, B. Stinner, The Cahn-Hilliard equation on an evolving surface, Preprint arXiv:1607.05627
[32] Pego, R. L., Front migration in the nonlinear Cahn-Hilliard equation, Proc. R. Soc. Lond. Ser. A, 422, 261-278 (1989) · Zbl 0701.35159
[33] Wu, H.; Zheng, S., Convergence to equilibrium for the Cahn-Hilliard equation with dynamic boundary conditions, J. Differential Equations, 204, 511-531 (2004) · Zbl 1068.35018
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