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Optimal control for the convective Cahn-Hilliard equation in 2D case. (English) Zbl 1298.49013

The Cahn-Hilliard model is known and has been intensively studied theoretically as well as in practical applications for more than a half century: [J. W. Cahn and J. E. Hilliard, “Free energy of a nonuniform system. I. Interfacial free energy”, J. Chem. Phys. 28, 258-267 (1958)]. In a previous paper [Appl. Anal. 92, No. 5, 1028–1045 (2013; Zbl 1269.49003)], the authors considered the optimal control problem for the 1D convective Cahn-Hilliard model with periodic boundary conditions. The present article is devoted to an investigation of the distributed control problem subject to the initial-boundary value problem of a 2D convective Cahn-Hilliard equation. The investigation follows the well known steps introduced by Lions in solving distributed optimal control problems. Under some additional assumptions on the model coefficients, the existence and uniqueness of a weak solution is proved first. Another result is connected to the optimality system and the existence of an optimal control.

MSC:

49J20 Existence theories for optimal control problems involving partial differential equations
49K20 Optimality conditions for problems involving partial differential equations
35K55 Nonlinear parabolic equations

Citations:

Zbl 1269.49003
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References:

[1] Hintermuller, M., Wegner, D.: Distributed optimal control of the Cahn-Hilliard system including the case of a double-obstacle homogeneous free energy density. SIAM J. Control Optim. 50, 388-418 (2012) · Zbl 1250.35131 · doi:10.1137/110824152
[2] Jeong, J.-M., Ju, E.-Y., Cheon, S.-J.: Optimal control problems for evolution equations of parabolic type with nonlinear perturbations. J. Optim. Theory Appl. 151, 573-588 (2011) · Zbl 1234.49023 · doi:10.1007/s10957-011-9866-7
[3] Lions, J.L.: Optimal Control of Systems Governed by Parttial Differential Equations. Springer, Berlin (1971) · doi:10.1007/978-3-642-65024-6
[4] Zhao, X.P., Liu, C.C.: Optimal control problem for viscous Cahn-Hilliard equation. Nonlinear Anal. 74, 6348-6357 (2011) · Zbl 1228.49008 · doi:10.1016/j.na.2011.06.015
[5] Yong, J., Zheng, S.: Feedback stabilization and optimal control for the Cahn-Hilliard equation. Nonlinear Anal. TMA 17, 431-444 (1991) · Zbl 0765.93067 · doi:10.1016/0362-546X(91)90138-Q
[6] Ryu, S.-U., Yagi, A.: Optimal control of Keller-Segel equations. J. Math. Anal. Appl. 256, 45-66 (2001) · Zbl 0982.49006 · doi:10.1006/jmaa.2000.7254
[7] Ryu, S.-U.: Optimal control problems governed by some semilinear parabolic equations. Nonlinear Anal. 56, 241-252 (2004) · Zbl 1054.49002 · doi:10.1016/j.na.2003.09.013
[8] Tian, L., Shen, C.: Optimal control of the viscous Degasperis-Procesi equation. J. Math. Phys. 48(11), 113513 (2007) · Zbl 1153.81442 · doi:10.1063/1.2804755
[9] Tian, L., Shen, C., Ding, D.: Optimal control of the viscous Camassa-Holm equation. Nonlinear Anal. RWA 10(1), 519-530 (2009) · Zbl 1154.49300 · doi:10.1016/j.nonrwa.2007.10.016
[10] 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 · doi:10.1063/1.1744102
[11] Novick-Cohen, A., Segel, L.A.: Nonlinear aspects of the Cahn-Hilliard equation. Phys. D 10, 277-298 (1984) · doi:10.1016/0167-2789(84)90180-5
[12] Novick-Cohen, A.: Energy methods for the Cahn-Hilliard equation. Q. Appl. Math. 46, 681-690 (1988) · Zbl 0685.35050
[13] Eden, A., Kalantarov, V.K.: 3D convective Cahn-Hilliard equation. Commun. Pure Appl. Anal. 6, 1075-1086 (2007) · Zbl 1140.35357 · doi:10.3934/cpaa.2007.6.1075
[14] Elliott, C.M., Zheng, S.M.: On the Cahn-Hilliard equation. Arch. Ration. Mech. Anal. 96, 339-357 (1986) · Zbl 0624.35048 · doi:10.1007/BF00251803
[15] Dlotko, T.: Global attractor for the Cahn-Hilliard equation in \[H^2\] H2 and \[H^3\] H3. J. Differ. Equ. 113, 381-393 (1994) · Zbl 0828.35015 · doi:10.1006/jdeq.1994.1129
[16] Temam, R.: Infinite Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, vol. 68. Springer, New York (1988) · Zbl 0662.35001 · doi:10.1007/978-1-4684-0313-8
[17] Debussche, A., Dettori, L.: On the Cahn-Hilliard equation with a logarithmic free energy. Nonlinear Anal. 24, 1491-1514 (1995) · Zbl 0831.35088 · doi:10.1016/0362-546X(94)00205-V
[18] Cherfils, L., Miranville, A., Zelik, S.: The Cahn-Hilliard equation with logarithmic potentials. Milan J. Math. 79, 561-596 (2011) · Zbl 1250.35129 · doi:10.1007/s00032-011-0165-4
[19] 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 · doi:10.3934/cpaa.2009.8.881
[20] Yin, J.X.: On the Cahn-Hilliard equation with nonlinear principal part. J. Partial Differ. Equ. 7, 77-96 (1994) · Zbl 0823.35036
[21] Huang, R., Yin, J.X., Wang, L.W.: Non-blow-up phenomenon for the Cahn-Hilliard equation with non-constant mobility. J. Math. Anal. Appl. 379, 58-64 (2011) · Zbl 1228.35065 · doi:10.1016/j.jmaa.2010.12.028
[22] Yin, J.X.: On the existence of nonnegative continuous solutions of the Cahn-Hilliard equation. J. Differ. Equ. 97, 310-327 (1992) · Zbl 0772.35051 · doi:10.1016/0022-0396(92)90075-X
[23] Elliott, C.M., Garcke, H.: On the Cahn-Hilliard equation with degenerate mobility. SIAM J. Math. Anal. 27, 404-423 (1996) · Zbl 0856.35071 · doi:10.1137/S0036141094267662
[24] Schimperna, G.: Global attractors for Cahn-Hilliard equations with nonconstant mobility. Nonlinearity 20, 2365-2387 (2007) · Zbl 1133.35024 · doi:10.1088/0951-7715/20/10/006
[25] Chen, X.F.: Global asymptotic limit of solutions of the Cahn-Hilliard equation. J. Differ. Geom. 44, 262-311 (1996) · Zbl 0874.35045
[26] 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) · Zbl 0861.35039 · doi:10.1017/S0956792500002369
[27] Shen, J., Yang, X.F.: Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discret. Contin. Dyn. Syst. 28, 1669-1691 (2010) · Zbl 1201.65184 · doi:10.3934/dcds.2010.28.1669
[28] Banas, L., Nurnberg, R.: Adaptive finite element methods for Cahn-Hilliard equations. J. Comp. Appl. Math. 218, 2-11 (2008) · Zbl 1143.65076 · doi:10.1016/j.cam.2007.04.030
[29] Golovin, A.A., Davis, S.H., Nepomnyashchy, A.A.: A convective Cahn-Hilliard model for the formation of facets and corners in crystal growth. Phys. D 122, 202-230 (1998) · Zbl 0952.74050 · doi:10.1016/S0167-2789(98)00181-X
[30] Watson, S.J., Otto, F., Rubinstein, B.Y., Davis, S.H.: Coarsening dynamics of the convective Cahn-Hilliard equation. Phys. D 178, 127-148 (2003) · Zbl 1011.76021 · doi:10.1016/S0167-2789(03)00048-4
[31] Zarks, M.A., Podolny, A., Nepomnyashchy, A.A., Golovin, A.A.: Periodic stationary patterns governed by a convective Cahn-Hilliard equation. SIAM J. Appl. Math. 66, 700-720 (2005) · Zbl 1102.35009 · doi:10.1137/040615766
[32] Eden, A., Kalantarov, V.K.: The convective Cahn-Hilliard equation. Appl. Math. Lett. 20, 455-461 (2007) · Zbl 1166.35368 · doi:10.1016/j.aml.2006.05.014
[33] Liu, C.C.: On the convective Cahn-Hillirad equation with degenerate mobility. J. Math. Anal. Appl. 344, 124-144 (2008) · Zbl 1158.35077 · doi:10.1016/j.jmaa.2008.02.027
[34] Zhao, X.P., Liu, B.: The existence of global attractor for convective Cahn-Hilliard equation. J. Korean Math. Soc. 49, 357-378 (2012) · Zbl 1242.35058 · doi:10.4134/JKMS.2012.49.2.357
[35] Zhao, X.P., Liu, C.C.: Optimal control of the convective Cahn-Hilliard equation. Appl. Anal. 92, 1028-1045 (2013) · Zbl 1269.49003 · doi:10.1080/00036811.2011.643786
[36] Simon, J.: Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure. SIAM J. Math. Anal. 21(5), 1093-1117 (1990) · Zbl 0702.76039 · doi:10.1137/0521061
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