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Existence and uniqueness of global weak solutions to a Cahn-Hilliard-Stokes-Darcy system for two phase incompressible flows in karstic geometry. (English) Zbl 1302.35217
Summary: We study the well-posedness of a coupled Cahn-Hilliard-Stokes-Darcy system which is a diffuse-interface model for essentially immiscible two phase incompressible flows with matched density in a karstic geometry. Existence of finite energy weak solution that is global in time is established in both 2D and 3D. Weak-strong uniqueness property of the weak solutions is provided as well.

MSC:
35K61 Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations
76S05 Flows in porous media; filtration; seepage
76D07 Stokes and related (Oseen, etc.) flows
35Q35 PDEs in connection with fluid mechanics
35D30 Weak solutions to PDEs
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References:
[1] Han, D.; Sun, D.; Wang, X., Two phase flows in karstic geometry, Math. Methods Appl. Sci., (2013)
[2] Çeşmelioğlu, A.; Rivière, B., Analysis of time-dependent Navier-Stokes flow coupled with Darcy flow, J. Numer. Math., 16, 249-280, (2008) · Zbl 1159.76010
[3] Badea, L.; Discacciati, M.; Quarteroni, A., Numerical analysis of the Navier-Stokes/Darcy coupling, Numer. Math., 115, 195-227, (2010) · Zbl 1423.35304
[4] Boyer, F., Mathematical study of multi-phase flow under shear through order parameter formulation, Asymptot. Anal., 20, 175-212, (1999) · Zbl 0937.35123
[5] Liu, C.; Shen, J., A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method, Phys. D, 179, 211-228, (2003) · Zbl 1092.76069
[6] Abels, H.; Lengeler, D., On sharp interface limits for diffuse interface models for two-phase flows, (2012)
[7] Lowengrub, J.; Truskinovsky, L., Quasi-incompressible Cahn-Hilliard fluids and topological transitions, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 454, 2617-2654, (1998) · Zbl 0927.76007
[8] Bear, J., Dynamics of fluids in porous media, (1988), Courier Dover Publications · Zbl 1191.76002
[9] Jones, I., Low Reynolds-number flow past a porous spherical shell, Proc. Cambridge Philos. Soc., 73, 231-238, (1973) · Zbl 0262.76061
[10] Saffman, P. G., On the boundary condition at the interface of a porous medium, Stud. Appl. Math., 1, 93-101, (1971) · Zbl 0271.76080
[11] Beavers, G.; Joseph, D., Boundary conditions at a naturally permeable wall, J. Fluid Mech., 30, 197-207, (1967)
[12] Jäger, W.; Mikelić, A., On the interface boundary condition of Beavers, Joseph, and Saffman, SIAM J. Appl. Math., 60, 1111-1127, (2000) · Zbl 0969.76088
[13] Discacciati, M.; Miglio, E.; Quarteroni, A., Mathematical and numerical models for coupling surface and groundwater flows, Appl. Numer. Math., 43, 57-74, (2002) · Zbl 1023.76048
[14] Discacciati, M.; Quarteroni, A., Analysis of a domain decomposition method for the coupling of the Stokes and Darcy equations, (Numerical Mathematics and Advanced Applications, vol. 320, (2003), Springer Milan), 3-20 · Zbl 1254.76051
[15] Discacciati, M.; Quarteroni, A., Navier-Stokes/Darcy coupling: modeling, analysis, and numerical approximation, Rev. Mat. Complut., 22, 315-426, (2009) · Zbl 1172.76050
[16] Layton, W. J.; Schieweck, F.; Yotov, I., Coupling fluid flow with porous media flow, SIAM J. Numer. Anal., 40, 2195-2218, (2002) · Zbl 1037.76014
[17] Cao, Y.; Gunzburger, M.; Hua, F.; Wang, X., Coupled Stokes-Darcy model with Beavers-Joseph interface boundary condition, Commun. Math. Sci., 8, 1-25, (2010) · Zbl 1189.35244
[18] Chen, N.; Gunzburger, M.; Wang, X., Asymptotic analysis of the differences between the Stokes-Darcy system with different interface conditions and the Stokes-Brinkman system, J. Math. Anal. Appl., 368, 658-676, (2010) · Zbl 1352.35093
[19] Cao, Y.; Gunzburger, M.; Hua, F.; Wang, X., Analysis and finite element approximation of a coupled, continuum pipe-flow/Darcy model for flow in porous media with embedded conduits, Numer. Methods Partial Differential Equations, 27, 1242-1252, (2011) · Zbl 1282.76172
[20] Chen, W.; Gunzburger, M.; Hua, F.; Wang, X., A parallel Robin-Robin domain decomposition method for the Stokes-Darcy system, SIAM J. Numer. Anal., 49, 1064-1084, (2011) · Zbl 1414.76017
[21] Chen, W.; Gunzburger, M.; Sun, D.; Wang, X., Efficient and long-time accurate second-order methods for the Stokes-Darcy system, SIAM J. Numer. Anal., 51, 2563-2584, (2013) · Zbl 1282.76094
[22] Chidyagwai, P.; Rivière, B., On the solution of the coupled Navier-Stokes and Darcy equations, Comput. Methods Appl. Mech. Engrg., 198, 3806-3820, (2009) · Zbl 1230.76023
[23] Çeşmelioğlu, A.; Rivière, B., Existence of a weak solution for the fully coupled Navier-Stokes/Darcy-transport problem, J. Differential Equations, 252, 4138-4175, (2012) · Zbl 1234.35178
[24] Çeşmelioğlu, A.; Girault, V.; Rivière, B., Time-dependent coupling of Navier-Stokes and Darcy flows, ESAIM: M2AN, 47, 539-554, (2013) · Zbl 1267.76096
[25] Deimling, K., Nonlinear functional analysis, (1985), Springer-Verlag Berlin · Zbl 0559.47040
[26] Abels, H., Existence of weak solutions for a diffuse interface model for viscous, incompressible fluids with general densities, Comm. Math. Phys., 289, 45-73, (2009) · Zbl 1165.76050
[27] Feng, X.; Wise, S., Analysis of a Darcy-Cahn-Hilliard diffuse interface model for the Hele-Shaw flow and its fully discrete finite element approximation, SIAM J. Numer. Anal., 50, 1320-1343, (2012) · Zbl 1426.76258
[28] Wise, S. M., Unconditionally stable finite difference, nonlinear multigrid simulation of the Cahn-Hilliard-Hele-Shaw system of equations, J. Sci. Comput., 44, 38-68, (2010) · Zbl 1203.76153
[29] Han, D.; Wang, X., A second order in time, uniquely solvable, unconditionally stable numerical scheme for Cahn-Hilliard-Navier-Stokes equation, (2014), eprint
[30] Abels, H., On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities, Arch. Ration. Mech. Anal., 194, 463-506, (2009) · Zbl 1254.76158
[31] Zhao, L.; Wu, H.; Huang, H., Convergence to equilibrium for a phase-field model for the mixture of two viscous incompressible fluids, Commun. Math. Sci., 7, 939-962, (2009) · Zbl 1183.35224
[32] Gal, C. G.; Grasselli, M., Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27, 401-436, (2010) · Zbl 1184.35055
[33] Wang, X.; Zhang, Z., Well-posedness of the Hele-Shaw-Cahn-Hilliard system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30, 367-384, (2013) · Zbl 1291.35240
[34] Wang, X.; Wu, H., Long-time behavior for the Hele-Shaw-Cahn-Hilliard system, Asymptot. Anal., 78, 217-245, (2012) · Zbl 1246.35164
[35] Lowengrub, J.; Titi, E.; Zhao, K., Analysis of a mixture model of tumor growth, European J. Appl. Math., 24, 691-734, (2013) · Zbl 1292.35153
[36] Diegel, A. E.; Feng, X.; Wise, S. M., Analysis of a mixed finite element method for a Cahn-Hilliard-Darcy-Stokes system, (2014)
[37] Triebel, H., Interpolation theory, function spaces, differential operators, (1978), North-Holland Amsterdam · Zbl 0387.46033
[38] Lions, J.-L.; Magenes, E., Non-homogeneous boundary value problems and applications, vol. I, Grundlehren Math. Wiss., vol. 181, (1972), Springer-Verlag New York, translated from the French by P. Kenneth · Zbl 0223.35039
[39] Grisvard, P., Elliptic problems in nonsmooth domains, Monogr. Stud. Math., vol. 24, (1985), Pitman (Advanced Publishing Program) Boston, MA · Zbl 0695.35060
[40] Lions, J.-L., Quelques methodes de resolution des provlémes aux limites non linéaires, (1969), Dunod Paris · Zbl 0189.40603
[41] Temam, R., Navier-Stokes equations. theory and numerical analysis, Stud. Math. Appl., vol. 2, (1977), North-Holland Amsterdam, New York, Oxford · Zbl 0383.35057
[42] Eyre, D. J., Unconditionally gradient stable time marching the Cahn-Hilliard equation, (Computational and Mathematical Models of Microstructural Evolution, San Francisco, CA, 1998, Mater. Res. Soc. Sympos. Proc., vol. 529, (1998), MRS Warrendale, PA), 39-46
[43] Horgan, C. O., Korn’s inequalities and their applications in continuum mechanics, SIAM Rev., 37, 491-511, (1995) · Zbl 0840.73010
[44] Girault, V.; Raviart, P.-A., Finite element methods for Navier-Stokes equations, Springer Ser. Comput. Math., vol. 5, (1986), Springer-Verlag Berlin
[45] Showalter, R. E., Monotone operators in Banach space and nonlinear partial differential equations, Math. Surveys Monogr., vol. 49, (1997), American Mathematical Society Providence, RI · Zbl 0870.35004
[46] Brézis, H.; Gallouet, T., Nonlinear Schrödinger evolution equations, Nonlinear Anal., 4, 677-681, (1980) · Zbl 0451.35023
[47] Simon, J., Compact sets in the space \(L^p(0, T; B)\), Ann. Mat. Pura Appl. (4), 146, 65-96, (1987) · Zbl 0629.46031
[48] Colli, P.; Krejčí, P.; Rocca, E.; Sprekels, J., Nonlinear evolution inclusions arising from phase change models, Czechoslovak Math. J., 57, 132, 1067-1098, (2007) · Zbl 1174.35021
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