# American Institute of Mathematical Sciences

March  2016, 15(2): 299-317. doi: 10.3934/cpaa.2016.15.299

## On the nonlocal Cahn-Hilliard-Brinkman and Cahn-Hilliard-Hele-Shaw systems

 1 Mathematical Institute, University of Oxford, Oxford OX2 6GG 2 Dipartimento di Matematica, Politecnico di Milano, Via E. Bonardi, 9, I-20133 Milano

Received  September 2015 Revised  November 2015 Published  January 2016

The phase separation of an isothermal incompressible binary fluid in a porous medium can be described by the so-called Brinkman equation coupled with a convective Cahn-Hilliard (CH) equation. The former governs the average fluid velocity $\mathbf{u}$, while the latter rules evolution of $\varphi$, the difference of the (relative) concentrations of the two phases. The two equations are known as the Cahn-Hilliard-Brinkman (CHB) system. In particular, the Brinkman equation is a Stokes-like equation with a forcing term (Korteweg force) which is proportional to $\mu\nabla\varphi$, where $\mu$ is the chemical potential. When the viscosity vanishes, then the system becomes the Cahn-Hilliard-Hele-Shaw (CHHS) system. Both systems have been studied from the theoretical and the numerical viewpoints. However, theoretical results on the CHHS system are still rather incomplete. For instance, uniqueness of weak solutions is unknown even in 2D. Here we replace the usual CH equation with its physically more relevant nonlocal version. This choice allows us to prove more about the corresponding nonlocal CHHS system. More precisely, we first study well-posedness for the CHB system, endowed with no-slip and no-flux boundary conditions. Then, existence of a weak solution to the CHHS system is obtained as a limit of solutions to the CHB system. Stronger assumptions on the initial datum allow us to prove uniqueness for the CHHS system. Further regularity properties are obtained by assuming additional, though reasonable, assumptions on the interaction kernel. By exploiting these properties, we provide an estimate for the difference between the solution to the CHB system and the one to the CHHS system with respect to viscosity.
Citation: Francesco Della Porta, Maurizio Grasselli. On the nonlocal Cahn-Hilliard-Brinkman and Cahn-Hilliard-Hele-Shaw systems. Communications on Pure and Applied Analysis, 2016, 15 (2) : 299-317. doi: 10.3934/cpaa.2016.15.299
##### References:
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Wise, An efficient, energy stable scheme for the Cahn-Hilliard-Brinkman system, Commun. Comput. Phys., 13 (2013), 929-957. [7] F. Della Porta and M. Grasselli, Convective nonlocal Cahn-Hilliard equations with reaction terms, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1529-1553. doi: 10.3934/dcdsb.2015.20.1529. [8] M. Dai, E. Feireisl, E. Rocca, G. Schimperna and M. E. Schonbeck, Analysis of a diffuse interface model of multispecies tumor growth, preprint, arXiv:1507.07683. [9] A. Diegel, X. Feng and S. Wise, Analysis of a mixed finite element method for a Cahn-Hilliard-Darcy-Stokes system, SIAM J. Numer. Anal., 53 (2015), 127-152. doi: 10.1137/130950628. [10] X. Feng and S. Wise, 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 (2012), 1320-1343. doi: 10.1137/110827119. [11] S. Frigeri and M. Grasselli, Global and trajectory attractors for a nonlocal Cahn-Hilliard-Navier-Stokes system, J. Dynam. Differential Equations, 24 (2012), 827-856. doi: 10.1007/s10884-012-9272-3. [12] S. Frigeri and M. Grasselli, Nonlocal Cahn-Hilliard-Navier-Stokes systems with singular potentials, Dyn. Partial Differ. Equ., 9 (2012), 273-304. doi: 10.4310/DPDE.2012.v9.n4.a1. [13] S. Frigeri, C. G. Gal and M. Grasselli, On nonlocal Cahn-Hilliard-Navier-Stokes systems in two dimensions, preprint, arXiv:1401.7954. [14] S. Frigeri, M. Grasselli and P. Krejčí, Strong solutions for two-dimensional nonlocal Cahn-Hilliard-Navier-Stokes systems, J. Differential Equations, 255 (2013), 2587-2614. doi: 10.1016/j.jde.2013.07.016. [15] S. Frigeri, M. Grasselli and E. Rocca, A diffusive interface model for two-phase incompressible flows with nonlocal interactions and nonconstant mobility, Nonlinearity, 28 (2015), 1257-1293. doi: 10.1088/0951-7715/28/5/1257. [16] S. Frigeri, E. Rocca and J. Sprekels, Optimal distributed control of a nonlocal Cahn-Hilliard/Navier-Stokes system in 2D, preprint, arXiv:1411.1627. [17] H. Gajewski and K. Zacharias, On a nonlocal phase separation model, J. Math. Anal. Appl., 286 (2003), 11-31. doi: 10.1016/S0022-247X(02)00425-0. [18] C. G. Gal and M. Grasselli, Longtime behavior of nonlocal Cahn-Hilliard equations, Discrete Contin. Dyn. Syst., 34 (2014), 145-179. doi: 10.3934/dcds.2014.34.145. [19] G. Giacomin and J. L. Lebowitz, Exact macroscopic description of phase segregation in model alloys with long range interactions, Phys. Rev. Lett., 76 (1996), 1094-1097. doi: 10.1103/PhysRevLett.76.1094. [20] G. Giacomin and J. L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits, J. Stat. Phys., 87 (1997), 37-61. doi: 10.1007/BF02181479. [21] G. Giacomin and J. L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions. II. Interface motion, SIAM J. Appl. Math., 58 (1998), 1707-1729. doi: 10.1137/S0036139996313046. [22] Z. Guan, J. S. Lowengrub, C. Wang and S. M. Wise, Second order convex splitting schemes for periodic nonlocal Cahn-Hilliard and Allen-Cahn equations, J. Comput. Phys., 277 (2014), 48-71. doi: 10.1016/j.jcp.2014.08.001. [23] Z. Guan, C. Wang and S. M. Wise, A convergent convex splitting scheme for the periodic nonlocal Cahn-Hilliard equation, Numer. Math., 128 (2014), 377-406. doi: 10.1007/s00211-014-0608-2. [24] S.-O. Londen and H. Petzeltová, Regularity and separation from potential barriers for a non-local phase-field system, J. Math. Anal. Appl., 379 (2011), 724-735. doi: 10.1016/j.jmaa.2011.02.003. [25] S.-O. Londen and H. Petzeltová, Convergence of solutions of a non-local phase-field system, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 653-670. doi: 10.3934/dcdss.2011.4.653. [26] J. Lowengrub, E. Titi and K. Zhao, Analysis of a mixture model of tumor growth, European J. Appl. Math., 24 (2013), 1-44. doi: 10.1017/S0956792513000144. [27] S. Melchionna and E. Rocca, On a nonlocal Cahn-Hilliard equation with a reaction term, Adv. Math. Sci. Appl., 24 (2014), 461-497. [28] W. Ngamsaad, J. Yojina and W. Triampo, Theoretical studies of phase-separation kinetics in a Brinkman porous medium, J. Phys. A, 43 (2010), 202001(7pp). [29] E. Rocca and J. Sprekels, Optimal distributed control of a nonlocal convective Cahn-Hilliard equation by the velocity in three dimensions, SIAM J. Control Optim., 53 (2015), 1654-1680. doi: 10.1137/140964308. [30] M. Schmuck, M. Pradas, G. A. Pavliotis and S. Kalliadasis, Derivation of effective macroscopic Stokes-Cahn-Hilliard equations for periodic immiscible flows in porous media, Nonlinearity, 26 (2013), 3259-3277. doi: 10.1088/0951-7715/26/12/3259. [31] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001. [32] X. Wang and H. Wu, Long-time behavior for the Hele-Shaw-Cahn-Hilliard system, Asymptot. Anal., 78 (2012), 217-245. [33] X. Wang and Z. Zhang, Well-posedness of the Hele-Shaw-Cahn-Hilliard system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 367-384. doi: 10.1016/j.anihpc.2012.06.003. [34] S. M. Wise, Unconditionally stable finite difference, nonlinear multigrid simulation of the Cahn-Hilliard-Hele-Shaw system of equations, J. Sci. Comput., 44 (2010), 38-68. doi: 10.1007/s10915-010-9363-4.

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##### References:
 [1] P. W. Bates and J. Han, The Neumann boundary problem for a nonlocal Cahn-Hilliard equation, J. Differential Equations, 212 (2005), 235-277. doi: 10.1016/j.jde.2004.07.003. [2] J. Bedrossian, N. Rodriguez and A. Bertozzi, Local and global well-posedness for an aggregation equation and Patlak-Keller-Segel models with degenerate diffusion, Nonlinearity, 24 (2011), 1683-1714. doi: 10.1088/0951-7715/24/6/001. [3] S. Bosia, M. Conti and M. Grasselli, On the Cahn-Hilliard-Brinkman System, Commun. Math. Sci., 13 (2015), 1541-1567. doi: 10.4310/CMS.2015.v13.n6.a9. [4] H. C. Brinkman, A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles, Appl. Sci. Res., A1 (1947), 27-36. [5] P. Colli, S. Frigeri and M. Grasselli, Global existence of weak solutions to a nonlocal Cahn-Hilliard-Navier-Stokes system, J. Math. Anal. Appl., 386 (2012), 428-444. doi: 10.1016/j.jmaa.2011.08.008. [6] C. Collins, J. Shen and S. M. Wise, An efficient, energy stable scheme for the Cahn-Hilliard-Brinkman system, Commun. Comput. Phys., 13 (2013), 929-957. [7] F. Della Porta and M. Grasselli, Convective nonlocal Cahn-Hilliard equations with reaction terms, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1529-1553. doi: 10.3934/dcdsb.2015.20.1529. [8] M. Dai, E. Feireisl, E. Rocca, G. Schimperna and M. E. Schonbeck, Analysis of a diffuse interface model of multispecies tumor growth, preprint, arXiv:1507.07683. [9] A. Diegel, X. Feng and S. Wise, Analysis of a mixed finite element method for a Cahn-Hilliard-Darcy-Stokes system, SIAM J. Numer. Anal., 53 (2015), 127-152. doi: 10.1137/130950628. [10] X. Feng and S. Wise, 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 (2012), 1320-1343. doi: 10.1137/110827119. [11] S. Frigeri and M. Grasselli, Global and trajectory attractors for a nonlocal Cahn-Hilliard-Navier-Stokes system, J. Dynam. Differential Equations, 24 (2012), 827-856. doi: 10.1007/s10884-012-9272-3. [12] S. Frigeri and M. Grasselli, Nonlocal Cahn-Hilliard-Navier-Stokes systems with singular potentials, Dyn. Partial Differ. Equ., 9 (2012), 273-304. doi: 10.4310/DPDE.2012.v9.n4.a1. [13] S. Frigeri, C. G. Gal and M. Grasselli, On nonlocal Cahn-Hilliard-Navier-Stokes systems in two dimensions, preprint, arXiv:1401.7954. [14] S. Frigeri, M. Grasselli and P. Krejčí, Strong solutions for two-dimensional nonlocal Cahn-Hilliard-Navier-Stokes systems, J. Differential Equations, 255 (2013), 2587-2614. doi: 10.1016/j.jde.2013.07.016. [15] S. Frigeri, M. Grasselli and E. Rocca, A diffusive interface model for two-phase incompressible flows with nonlocal interactions and nonconstant mobility, Nonlinearity, 28 (2015), 1257-1293. doi: 10.1088/0951-7715/28/5/1257. [16] S. Frigeri, E. Rocca and J. Sprekels, Optimal distributed control of a nonlocal Cahn-Hilliard/Navier-Stokes system in 2D, preprint, arXiv:1411.1627. [17] H. Gajewski and K. Zacharias, On a nonlocal phase separation model, J. Math. Anal. Appl., 286 (2003), 11-31. doi: 10.1016/S0022-247X(02)00425-0. [18] C. G. Gal and M. Grasselli, Longtime behavior of nonlocal Cahn-Hilliard equations, Discrete Contin. Dyn. Syst., 34 (2014), 145-179. doi: 10.3934/dcds.2014.34.145. [19] G. Giacomin and J. L. Lebowitz, Exact macroscopic description of phase segregation in model alloys with long range interactions, Phys. Rev. Lett., 76 (1996), 1094-1097. doi: 10.1103/PhysRevLett.76.1094. [20] G. Giacomin and J. L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits, J. Stat. Phys., 87 (1997), 37-61. doi: 10.1007/BF02181479. [21] G. Giacomin and J. L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions. II. Interface motion, SIAM J. Appl. Math., 58 (1998), 1707-1729. doi: 10.1137/S0036139996313046. [22] Z. Guan, J. S. Lowengrub, C. Wang and S. M. Wise, Second order convex splitting schemes for periodic nonlocal Cahn-Hilliard and Allen-Cahn equations, J. Comput. Phys., 277 (2014), 48-71. doi: 10.1016/j.jcp.2014.08.001. [23] Z. Guan, C. Wang and S. M. Wise, A convergent convex splitting scheme for the periodic nonlocal Cahn-Hilliard equation, Numer. Math., 128 (2014), 377-406. doi: 10.1007/s00211-014-0608-2. [24] S.-O. Londen and H. Petzeltová, Regularity and separation from potential barriers for a non-local phase-field system, J. Math. Anal. Appl., 379 (2011), 724-735. doi: 10.1016/j.jmaa.2011.02.003. [25] S.-O. Londen and H. Petzeltová, Convergence of solutions of a non-local phase-field system, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 653-670. doi: 10.3934/dcdss.2011.4.653. [26] J. Lowengrub, E. Titi and K. Zhao, Analysis of a mixture model of tumor growth, European J. Appl. Math., 24 (2013), 1-44. doi: 10.1017/S0956792513000144. [27] S. Melchionna and E. Rocca, On a nonlocal Cahn-Hilliard equation with a reaction term, Adv. Math. Sci. Appl., 24 (2014), 461-497. [28] W. Ngamsaad, J. Yojina and W. Triampo, Theoretical studies of phase-separation kinetics in a Brinkman porous medium, J. Phys. A, 43 (2010), 202001(7pp). [29] E. Rocca and J. Sprekels, Optimal distributed control of a nonlocal convective Cahn-Hilliard equation by the velocity in three dimensions, SIAM J. Control Optim., 53 (2015), 1654-1680. doi: 10.1137/140964308. [30] M. Schmuck, M. Pradas, G. A. Pavliotis and S. Kalliadasis, Derivation of effective macroscopic Stokes-Cahn-Hilliard equations for periodic immiscible flows in porous media, Nonlinearity, 26 (2013), 3259-3277. doi: 10.1088/0951-7715/26/12/3259. [31] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001. [32] X. Wang and H. Wu, Long-time behavior for the Hele-Shaw-Cahn-Hilliard system, Asymptot. Anal., 78 (2012), 217-245. [33] X. Wang and Z. Zhang, Well-posedness of the Hele-Shaw-Cahn-Hilliard system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 367-384. doi: 10.1016/j.anihpc.2012.06.003. [34] S. M. Wise, Unconditionally stable finite difference, nonlinear multigrid simulation of the Cahn-Hilliard-Hele-Shaw system of equations, J. Sci. Comput., 44 (2010), 38-68. doi: 10.1007/s10915-010-9363-4.
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