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Keywords:
Cahn-Hilliard; anisotropic behavior; gradient flow; curve of maximal slope; entropy
Summary:
We show existence of solutions to two types of generalized anisotropic Cahn-Hilliard problems: In the first case, we assume the mobility to be dependent on the concentration and its gradient, where the system is supplied with dynamic boundary conditions. In the second case, we deal with classical no-flux boundary conditions where the mobility depends on concentration $u$, gradient of concentration $\nabla u$ and the chemical potential $\Delta u-s'(u)$. The existence is shown using a newly developed generalization of gradient flows by the author and the theory of Young measures.
References:
[1] Abels, H., Wilke, M.: Convergence to equilibrium for the Cahn-Hilliard equation with a logarithmic free energy. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 67 3176-3193 (2007). DOI 10.1016/j.na.2006.10.002 | MR 2347608 | Zbl 1121.35018
[2] Adams, R. A.: Sobolev Spaces. Pure and Applied Mathematics 65. A Series of Monographs and Textbooks Academic Press, New York (1975). MR 0450957 | Zbl 0314.46030
[3] Brokate, M., Kenmochi, N., Müller, I., Rodriguez, J. F., Verdi, C.: Phase transitions and hysteresis. Lectures Given at the Third C.I.M.E., 1993, Montecatini Terme, Italy Lecture Notes in Mathematics 1584 Springer, Berlin (1994), A. Visintin. MR 1321829
[4] Buscaglia, G. C., Ausas, R. F.: Variational formulations for surface tension, capillarity and wetting. Comput. Methods Appl. Mech. Eng. 200 3011-3025 (2011). DOI 10.1016/j.cma.2011.06.002 | MR 2844033 | Zbl 1230.76047
[5] Cahn, J. W., Elliott, C. M., Novick-Cohen, A.: The Cahn-Hilliard equation with a concentration dependent mobility: motion by minus the Laplacian of the mean curvature. Eur. J. Appl. Math. 7 287-301 (1996). DOI 10.1017/S0956792500002369 | MR 1401172 | Zbl 0861.35039
[6] Changchun, L.: Cahn-Hilliard equation with terms of lower order and non-constant mobility. Electron. J. Qual. Theory Differ. Equ. 2003 (2003), 9 pp. (electronic only). MR 1986908 | Zbl 1032.35076
[7] Passo, R. Dal, Giacomelli, L., Novick-Cohen, A.: Existence for an Allen-Cahn/Cahn-Hilliard system with degenerate mobility. Interfaces Free Bound. 1 199-226 (1999). DOI 10.4171/IFB/9 | MR 1867131
[8] Debussche, A., Dettori, L.: On the Cahn-Hilliard equation with a logarithmic free energy. Nonlinear Anal., Theory Methods Appl. 24 1491-1514 (1995). DOI 10.1016/0362-546X(94)00205-V | MR 1327930 | Zbl 0831.35088
[9] Dellacherie, C., Meyer, P.-A.: Probabilities and Potential. North-Holland Mathematics Studies 29 North-Holland Publishing Company, Amsterdam (1978). MR 0521810 | Zbl 0494.60001
[10] Elliott, C. M., Garcke, H.: On the Cahn-Hilliard equation with degenerate mobility. SIAM J. Math. Anal. 27 404-423 (1996). DOI 10.1137/S0036141094267662 | MR 1377481 | Zbl 0856.35071
[11] Elliott, C. M., Zheng, S.: On the Cahn-Hilliard equation. Arch. Ration. Mech. Anal. 96 339-357 (1986). DOI 10.1007/BF00251803 | MR 0855754 | Zbl 0624.35048
[12] Gal, C. G.: Global well-posedness for the non-isothermal Cahn-Hilliard equation with dynamic boundary conditions. Adv. Differ. Equ. 12 1241-1274 (2007). MR 2372239 | Zbl 1162.35386
[13] 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). DOI 10.3934/cpaa.2009.8.881 | MR 2476663 | Zbl 1172.35417
[14] Grasselli, M., Miranville, A., Rossi, R., Schimperna, G.: Analysis of the Cahn-Hilliard equation with a chemical potential dependent mobility. Commun. Partial Differ. Equations 36 1193-1238 (2011). DOI 10.1080/03605302.2010.543945 | MR 2810586 | Zbl 1241.35024
[15] Heida, M.: Modeling Multiphase Flow in Porous Media with an Application to Permafrost Soil. Univ. Heidelberg, Naturwissenschaftlich-Mathematische Gesamtfakultät, Heidelberg; Charles Univ. Praha, Faculty of Mathematics and Physics (PhD Thesis), Praha (2011).
[16] Heida, M.: On the derivation of thermodynamically consistent boundary conditions for the Cahn-Hilliard-Navier-Stokes system. Internat. J. Engrg. Sci. 62 126-156 (2013). DOI 10.1016/j.ijengsci.2012.09.005 | MR 2996309
[17] Heida, M.: On systems of Cahn-Hilliard and Allen-Cahn equations considered as gradient flows in Hilbert spaces. J. Math. Anal. Appl. 423 410-455 (2015). DOI 10.1016/j.jmaa.2014.09.046 | MR 3273188
[18] Ito, A., Kenmochi, N., Niezgódka, M.: Large-time behaviour of non-isothermal models for phase separation. Proc. Conf. Elliptic and Parabolic Problems, 1994 Pitman Res. Notes Math. Ser. 325 Longman Scientific & Technical, Harlow 120-151 (1995). MR 1416579 | Zbl 0838.35053
[19] Lisini, S., Matthes, D., Savaré, G.: Cahn-Hilliard and thin film equations with nonlinear mobility as gradient flows in weighted-Wasserstein metrics. J. Differ. Equations 253 814-850 (2012). DOI 10.1016/j.jde.2012.04.004 | MR 2921215 | Zbl 1248.35095
[20] Liu, C.: On the convective Cahn-Hilliard equation with degenerate mobility. J. Math. Anal. Appl. 344 124-144 (2008). DOI 10.1016/j.jmaa.2008.02.027 | MR 2416296 | Zbl 1158.35077
[21] Liu, C., Qi, Y., Yin, J.: Regularity of solutions of the Cahn-Hilliard equation with non-constant mobility. Acta Math. Sin., Engl. Ser. 22 1139-1150 (2006). DOI 10.1007/s10114-005-0711-5 | MR 2245245 | Zbl 1106.35011
[22] Lowengrub, J. S., Rätz, A., Voigt, A.: Phase-field modeling of the dynamics of multicomponent vesicles: Spinodal decomposition, coarsening, budding, and fission. Phys. Rev. E 79 (2009), 031926, 13 pp. DOI 10.1103/PhysRevE.79.031926 | MR 2497179
[23] Mercker, M.: Models, numerics and simulations of deforming biological surfaces. Univ. Heidelberg, Naturwissenschaftlich-Mathematische Gesamtfakultät (PhD Thesis), Heidelberg (2012). Zbl 1304.92023
[24] Mercker, M., Marciniak-Czochra, A., Richter, T., Hartmann, D.: Modeling and computing of deformation dynamics of inhomogeneous biological surfaces. SIAM J. Appl. Math. 73 1768-1792 (2013). DOI 10.1137/120885553 | MR 3095724 | Zbl 1282.74015
[25] Mercker, M., Richter, T., Hartmann, D.: Sorting mechanisms and communication in phase separating coupled monolayers. J. Phys. Chem. B 115 11739-11745, DOI: 10.1021/jp204127g (2011). DOI 10.1021/jp204127g
[26] Miranville, A.: Existence of solutions for Cahn-Hilliard type equations. Discrete Contin. Dyn. Syst. 2003 Suppl. Vol., 630-637 (2003). MR 2018168 | Zbl 1070.35002
[27] Miranville, A., Zelik, S.: Robust exponential attractors for Cahn-Hilliard type equations with singular potentials. Math. Methods Appl. Sci. 27 545-582 (2004). DOI 10.1002/mma.464 | MR 2041814 | Zbl 1050.35113
[28] Miranville, A., Zelik, S.: The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions. Discrete Contin. Dyn. Syst. 28 275-310 (2010). DOI 10.3934/dcds.2010.28.275 | MR 2629483 | Zbl 1203.35046
[29] Mugnai, L., Röger, M.: The Allen-Cahn action functional in higher dimensions. Interfaces Free Bound. 10 45-78 (2008). DOI 10.4171/IFB/179 | MR 2383536 | Zbl 1288.93096
[30] Mugnai, L., Röger, M.: Convergence of perturbed Allen-Cahn equations to forced mean curvature flow. Indiana Univ. Math. J. 60 41-76 (2011). DOI 10.1512/iumj.2011.60.3949 | MR 2952409
[31] Novick-Cohen, A.: The Cahn-Hilliard equation. Handbook of Differential Equations: Evolutionary Equations. Vol. IV Elsevier/North-Holland, Amsterdam 201-228 (2008). MR 2508166 | Zbl 1185.35001
[32] Novick-Cohen, A.: The Cahn-Hilliard Equation: From Backwards Diffusion to Surface Diffusion. (to appear) in Cambridge University Press. MR 2508166
[33] Qian, T., Wang, X.-P., Sheng, P.: A variational approach to moving contact line hydrodynamics. J. Fluid Mech. 564 333-360 (2006). DOI 10.1017/S0022112006001935 | MR 2261865 | Zbl 1178.76296
[34] Racke, R., Zheng, S.: The Cahn-Hilliard equation with dynamic boundary conditions. Adv. Differ. Equ. 8 83-110 (2003). MR 1946559 | Zbl 1035.35050
[35] Röger, M., Schätzle, R.: On a modified conjecture of De Giorgi. Math. Z. 254 675-714 (2006). DOI 10.1007/s00209-006-0002-6 | MR 2253464 | Zbl 1126.49010
[36] Roidos, N., Schrohe, E.: The Cahn-Hilliard equation and the Allen-Cahn equation on manifolds with conical singularities. Commun. Partial Differ. Equations 38 925-943 (2013). DOI 10.1080/03605302.2012.736913 | MR 3046298 | Zbl 1272.58013
[37] Rossi, R.: On two classes of generalized viscous Cahn-Hilliard equations. Commun. Pure Appl. Anal. 4 405-430 (2005). DOI 10.3934/cpaa.2005.4.405 | MR 2149524 | Zbl 1078.35058
[38] Rossi, R., Savaré, G.: Gradient flows of non convex functionals in Hilbert spaces and applications. ESAIM, Control Optim. Calc. Var. 12 564-614 (2006). DOI 10.1051/cocv:2006013 | MR 2224826 | Zbl 1116.34048
[39] Serfaty, S.: Gamma-convergence of gradient flows on Hilbert and metric spaces and applications. Discrete Contin. Dyn. Syst. 31 1427-1451 (2011). MR 2836361 | Zbl 1239.35015
[40] Stefanelli, U.: The Brezis-Ekeland principle for doubly nonlinear equations. SIAM J. Control Optim. 47 1615-1642 (2008). DOI 10.1137/070684574 | MR 2425653 | Zbl 1194.35214
[41] Taylor, J. E., Cahn, J. W.: Linking anisotropic sharp and diffuse surface motion laws via gradient flows. J. Stat. Phys. 77 183-197 (1994). DOI 10.1007/BF02186838 | MR 1300532 | Zbl 0844.35044
[42] Temam, R.: Navier-Stokes Equations. Theory and Numerical Analysis. American Mathematical Society, Providence (2001). MR 1846644 | Zbl 0981.35001
[43] Torabi, S., Lowengrub, J., Voigt, A., Wise, S.: A new phase-field model for strongly anisotropic systems. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 465 1337-1359 (2009). DOI 10.1098/rspa.2008.0385 | MR 2500806 | Zbl 1186.80014
[44] Torabi, S., Wise, S., Lowengrub, J., Rätz, A., Voigt, A.: A new method for simulating strongly anisotropic Cahn-Hilliard equations. Materials Science and Technology-Association for Iron and Steel Technology 3 1432-1444 (2007).
[45] Wise, S., Kim, J., Lowengrub, J.: Solving the regularized, strongly anisotropic Cahn-Hilliard equation by an adaptive nonlinear multigrid method. J. Comput. Phys. 226 414-446 (2007). DOI 10.1016/j.jcp.2007.04.020 | MR 2356365 | Zbl 1310.82044
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