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The Cahn-Hilliard-Hele-Shaw system with singular potential. (English) Zbl 1394.35356
Summary: The Cahn-Hilliard-Hele-Shaw system is a fundamental diffuse-interface model for an incompressible binary fluid confined in a Hele-Shaw cell. It consists of a convective Cahn-Hilliard equation in which the velocity $$u$$ is subject to a Korteweg force through Darcy’s equation. In this paper, we aim to investigate the system with a physically relevant potential (i.e., of logarithmic type). This choice ensures that the (relative) concentration difference $$\phi$$ takes values within the admissible range. To the best of our knowledge, essentially all the available contributions in the literature are concerned with a regular approximation of the singular potential. Here, we first prove the existence of a global weak solution with finite energy that satisfies an energy dissipative property. Then in dimension two, we further obtain the uniqueness and regularity of global weak solutions. In particular, we show that any two-dimensional weak solution satisfies the so-called strict separation property, namely, if $$\varphi$$ is not a pure state at some initial time, then it stays instantaneously away from the pure states. When the spatial dimension is three, we prove the existence of a unique global strong solution, provided that the initial datum is regular enough and sufficiently close to any local minimizer of the free energy. This also yields the local Lyapunov stability of the local minimizer itself. Finally, we prove that under suitable assumptions any global solution converges to a single equilibrium as time goes to infinity.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 35B40 Asymptotic behavior of solutions to PDEs 35B41 Attractors 76D27 Other free boundary flows; Hele-Shaw flows 35D30 Weak solutions to PDEs 35D35 Strong solutions to PDEs 76S05 Flows in porous media; filtration; seepage 35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness 35A01 Existence problems for PDEs: global existence, local existence, non-existence
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##### References:
 [1] 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 [2] Abels, H.; Wilke, M., Convergence to equilibrium for the Cahn-Hilliard equation with a logarithmic free energy, Nonlinear Anal., 67, 3176-3193, (2007) · Zbl 1121.35018 [3] Amann, H., Compact embedding of vector-valued Sobolev and Besov spaces, Glas. Mat. Ser. III, 35, 161-177, (2000) · Zbl 0997.46029 [4] Ball, J. M., Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equation, J. Nonlinear Sci., J. Nonlinear Sci., 8, 233-502, (1998), Erratum · Zbl 0903.58020 [5] Bosia, S.; Conti, M.; Grasselli, M., On the Cahn-Hilliard-Brinkman system, Commun. Math. Sci., 13, 1541-1567, (2015) · Zbl 1330.35313 [6] Brézis, H., Functional analysis, Sobolev spaces and partial differential equations, Universitext, (2011), Springer New York · Zbl 1220.46002 [7] Cahn, J. W.; Hilliard, J. E., Free energy of a nonuniform system I. interfacial free energy, J. Chem. Phys., 2, 258-267, (1958) [8] Cazenave, T.; Haraux, A., An introduction to semilinear evolution equations, Oxford Lecture Series in Mathematics and Its Applications, vol. 13, (1998), The Clarendon Press, Oxford University Press New York · Zbl 0926.35049 [9] M. Conti, A. Giorgini, On the Cahn-Hilliard-Brinkman system with singular potential and nonconstant viscosity, preprint, 2016. [10] Dai, M.; Feireisl, E.; Rocca, E.; Schimperna, G.; Schonbek, M. E., Analysis of a diffuse interface model of multispecies tumor growth, Nonlinearity, 30, 1639-1658, (2017) · Zbl 1367.35185 [11] L. Dedé, H. Garcke, K.F. Lam, A Hele-Shaw-Cahn-Hilliard model for incompressible two-phase flows with different densities, MOX-Report No. 04/2017. [12] Della Porta, F.; Grasselli, M., On the nonlocal Cahn-Hilliard-Brinkman and Cahn-Hilliard-Hele-Shaw systems, Commun. Pure Appl. Anal., Commun. Pure Appl. Anal., 16, 369-372, (2017), Erratum · Zbl 06666338 [13] Fei, M., Global sharp interface limit of the Hele-Shaw-Cahn-Hilliard system, Math. Methods Appl. Sci., 40, 833-852, (2017) · Zbl 1359.35151 [14] 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 [15] Frigeri, S.; Grasselli, M., Nonlocal Cahn-Hilliard-Navier-Stokes systems with singular potential, Dyn. Partial Differ. Equ., 24, 827-856, (2012) · Zbl 1261.35105 [16] Gal, C. G.; Giorgini, A.; Grasselli, M., The nonlocal Cahn-Hilliard equation with singular potential: well-posedness, regularity and strict separation property, J. Differ. Equ., 263, 5253-5297, (2017) · Zbl 1400.35178 [17] Garcke, H.; Lam, K. F., Global weak solutions and asymptotic limits of a Cahn-hiliard-Darcy system modelling tumour growth, AIMS Math., 1, 318-360, (2016) [18] Giga, Y.; Miyakawa, T., Solutions in $$L^r$$ of the Navier-Stokes initial value problem, Arch. Ration. Mech. Anal., 89, 267-281, (1985) · Zbl 0587.35078 [19] Giorgini, A.; Grasselli, M.; Miranville, A., The Cahn-hiliard-oono equation with singular potential, Math. Models Methods Appl. Sci., 27, 2485-2510, (2017) · Zbl 1386.35023 [20] Girault, V.; Raviart, P. A., Finite element methods for Navier-Stokes equations. theory and algorithms, Springer Series in Computational Mathematics, vol. 5, (1986), Springer-Verlag Berlin [21] Han, D.; Wang, X.; Wu, H., Existence and uniqueness of global weak solutions to a Cahn-Hilliard-Stokes-Darcy system for two phase incompressible flows in karstic geometry, J. Differ. Equ., 257, 3887-3933, (2014) · Zbl 1302.35217 [22] Hohenberg, P. C.; Halperin, B. I., Theory of dynamic critical phenomena, Rev. Mod. Phys., 49, 435-479, (1977) [23] Jiang, J.; Wu, H.; Zheng, S., Well-posedness and long-time behavior of a non-autonomous Cahn-Hilliard-Darcy system with mass source modeling tumor growth, J. Differ. Equ., 259, 3032-3077, (2015) · Zbl 1330.35039 [24] Kenmochi, N.; Niezgódka, M.; Pawlow, I., Subdifferential operator approach to the Cahn-Hilliard equation with constraint, J. Differ. Equ., 117, 320-356, (1995) · Zbl 0823.35073 [25] Lee, H.-G.; Lowengrub, J.-S.; Goodman, J., Modeling pinch-off and reconnection in a Hele-Shaw cell. I. the models and their calibration, Phys. Fluids, 14, 492-512, (2002) · Zbl 1184.76316 [26] Lowengrub, J.; Titi, E.; Zhao, K., Analysis of a mixture model of tumor growth, Eur. J. Appl. Math., 24, 691-734, (2013) · Zbl 1292.35153 [27] Melchionna, S.; Rocca, E., Varifold solutions of a sharp interface limit of a diffuse interface model for tumor growth · Zbl 1390.35431 [28] 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 [29] Miranville, A.; Zelik, S., Attractors for dissipative partial differential equations in bounded and unbounded domains, (Dafermos, C. M.; Pokorny, M., Handbook of Differential Equations, Evolutionary Partial Differential Equations, vol. 4, (2008), Elsevier Amsterdam), 103-200 · Zbl 1221.37158 [30] Rocca, E.; Schimperna, G., Universal attractor for some singular phase transition systems, Physica D, 192, 279-307, (2004) · Zbl 1062.82015 [31] Rybka, P.; Hoffmann, K.-H., Convergence of solutions to Cahn-Hilliard equation, Commun. Partial Differ. Equ., 24, 1055-1077, (1999) · Zbl 0936.35032 [32] Simon, L., Asymptotics for a class of nonlinear evolution equation with applications to geometric problems, Ann. Math., 118, 525-571, (1983) · Zbl 0549.35071 [33] Temam, R., Navier-Stokes equations: theory and numerical analysis, (2001), AMS Providence · Zbl 0981.35001 [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] 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 [36] Wise, S. M., Unconditionally stable finite difference, nonlinear multigrid simulations of the Cahn-Hilliard-Hele-Shaw system of equations, J. Sci. Comput., 44, 38-68, (2010) · Zbl 1203.76153
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