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 & Applied Analysis, 2016, 15 (2) : 299-317. doi: 10.3934/cpaa.2016.15.299
References:
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J. Lowengrub, E. Titi and K. Zhao, Analysis of a mixture model of tumor growth,, \emph{European J. Appl. Math.}, 24 (2013), 1.  doi: 10.1017/S0956792513000144.  Google Scholar

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W. Ngamsaad, J. Yojina and W. Triampo, Theoretical studies of phase-separation kinetics in a Brinkman porous medium,, \emph{J. Phys. A}, 43 (2010).   Google Scholar

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X. Wang and H. Wu, Long-time behavior for the Hele-Shaw-Cahn-Hilliard system,, \emph{Asymptot. Anal.}, 78 (2012), 217.   Google Scholar

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X. Wang and Z. Zhang, Well-posedness of the Hele-Shaw-Cahn-Hilliard system,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 30 (2013), 367.  doi: 10.1016/j.anihpc.2012.06.003.  Google Scholar

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show all references

References:
[1]

P. W. Bates and J. Han, The Neumann boundary problem for a nonlocal Cahn-Hilliard equation,, \emph{J. Differential Equations}, 212 (2005), 235.  doi: 10.1016/j.jde.2004.07.003.  Google Scholar

[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,, \emph{Nonlinearity}, 24 (2011), 1683.  doi: 10.1088/0951-7715/24/6/001.  Google Scholar

[3]

S. Bosia, M. Conti and M. Grasselli, On the Cahn-Hilliard-Brinkman System,, \emph{Commun. Math. Sci.}, 13 (2015), 1541.  doi: 10.4310/CMS.2015.v13.n6.a9.  Google Scholar

[4]

H. C. Brinkman, A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles,, \emph{Appl. Sci. Res.}, A1 (1947), 27.   Google Scholar

[5]

P. Colli, S. Frigeri and M. Grasselli, Global existence of weak solutions to a nonlocal Cahn-Hilliard-Navier-Stokes system,, \emph{J. Math. Anal. Appl.}, 386 (2012), 428.  doi: 10.1016/j.jmaa.2011.08.008.  Google Scholar

[6]

C. Collins, J. Shen and S. M. Wise, An efficient, energy stable scheme for the Cahn-Hilliard-Brinkman system,, \emph{Commun. Comput. Phys.}, 13 (2013), 929.   Google Scholar

[7]

F. Della Porta and M. Grasselli, Convective nonlocal Cahn-Hilliard equations with reaction terms,, \emph{Discrete Contin. Dyn. Syst. Ser. B}, 20 (2015), 1529.  doi: 10.3934/dcdsb.2015.20.1529.  Google Scholar

[8]

M. Dai, E. Feireisl, E. Rocca, G. Schimperna and M. E. Schonbeck, Analysis of a diffuse interface model of multispecies tumor growth,, preprint, ().   Google Scholar

[9]

A. Diegel, X. Feng and S. Wise, Analysis of a mixed finite element method for a Cahn-Hilliard-Darcy-Stokes system,, \emph{SIAM J. Numer. Anal.}, 53 (2015), 127.  doi: 10.1137/130950628.  Google Scholar

[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,, \emph{SIAM J. Numer. Anal.}, 50 (2012), 1320.  doi: 10.1137/110827119.  Google Scholar

[11]

S. Frigeri and M. Grasselli, Global and trajectory attractors for a nonlocal Cahn-Hilliard-Navier-Stokes system,, \emph{J. Dynam. Differential Equations}, 24 (2012), 827.  doi: 10.1007/s10884-012-9272-3.  Google Scholar

[12]

S. Frigeri and M. Grasselli, Nonlocal Cahn-Hilliard-Navier-Stokes systems with singular potentials,, \emph{Dyn. Partial Differ. Equ.}, 9 (2012), 273.  doi: 10.4310/DPDE.2012.v9.n4.a1.  Google Scholar

[13]

S. Frigeri, C. G. Gal and M. Grasselli, On nonlocal Cahn-Hilliard-Navier-Stokes systems in two dimensions,, preprint, ().   Google Scholar

[14]

S. Frigeri, M. Grasselli and P. Krejčí, Strong solutions for two-dimensional nonlocal Cahn-Hilliard-Navier-Stokes systems,, \emph{J. Differential Equations}, 255 (2013), 2587.  doi: 10.1016/j.jde.2013.07.016.  Google Scholar

[15]

S. Frigeri, M. Grasselli and E. Rocca, A diffusive interface model for two-phase incompressible flows with nonlocal interactions and nonconstant mobility,, \emph{Nonlinearity}, 28 (2015), 1257.  doi: 10.1088/0951-7715/28/5/1257.  Google Scholar

[16]

S. Frigeri, E. Rocca and J. Sprekels, Optimal distributed control of a nonlocal Cahn-Hilliard/Navier-Stokes system in 2D,, preprint, ().   Google Scholar

[17]

H. Gajewski and K. Zacharias, On a nonlocal phase separation model,, \emph{J. Math. Anal. Appl.}, 286 (2003), 11.  doi: 10.1016/S0022-247X(02)00425-0.  Google Scholar

[18]

C. G. Gal and M. Grasselli, Longtime behavior of nonlocal Cahn-Hilliard equations,, \emph{Discrete Contin. Dyn. Syst.}, 34 (2014), 145.  doi: 10.3934/dcds.2014.34.145.  Google Scholar

[19]

G. Giacomin and J. L. Lebowitz, Exact macroscopic description of phase segregation in model alloys with long range interactions,, \emph{Phys. Rev. Lett.}, 76 (1996), 1094.  doi: 10.1103/PhysRevLett.76.1094.  Google Scholar

[20]

G. Giacomin and J. L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits,, \emph{J. Stat. Phys.}, 87 (1997), 37.  doi: 10.1007/BF02181479.  Google Scholar

[21]

G. Giacomin and J. L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions. II. Interface motion,, \emph{SIAM J. Appl. Math.}, 58 (1998), 1707.  doi: 10.1137/S0036139996313046.  Google Scholar

[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,, \emph{J. Comput. Phys.}, 277 (2014), 48.  doi: 10.1016/j.jcp.2014.08.001.  Google Scholar

[23]

Z. Guan, C. Wang and S. M. Wise, A convergent convex splitting scheme for the periodic nonlocal Cahn-Hilliard equation,, \emph{Numer. Math.}, 128 (2014), 377.  doi: 10.1007/s00211-014-0608-2.  Google Scholar

[24]

S.-O. Londen and H. Petzeltová, Regularity and separation from potential barriers for a non-local phase-field system,, \emph{J. Math. Anal. Appl.}, 379 (2011), 724.  doi: 10.1016/j.jmaa.2011.02.003.  Google Scholar

[25]

S.-O. Londen and H. Petzeltová, Convergence of solutions of a non-local phase-field system,, \emph{Discrete Contin. Dyn. Syst. Ser. S}, 4 (2011), 653.  doi: 10.3934/dcdss.2011.4.653.  Google Scholar

[26]

J. Lowengrub, E. Titi and K. Zhao, Analysis of a mixture model of tumor growth,, \emph{European J. Appl. Math.}, 24 (2013), 1.  doi: 10.1017/S0956792513000144.  Google Scholar

[27]

S. Melchionna and E. Rocca, On a nonlocal Cahn-Hilliard equation with a reaction term,, \emph{Adv. Math. Sci. Appl.}, 24 (2014), 461.   Google Scholar

[28]

W. Ngamsaad, J. Yojina and W. Triampo, Theoretical studies of phase-separation kinetics in a Brinkman porous medium,, \emph{J. Phys. A}, 43 (2010).   Google Scholar

[29]

E. Rocca and J. Sprekels, Optimal distributed control of a nonlocal convective Cahn-Hilliard equation by the velocity in three dimensions,, \emph{SIAM J. Control Optim.}, 53 (2015), 1654.  doi: 10.1137/140964308.  Google Scholar

[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,, \emph{Nonlinearity}, 26 (2013), 3259.  doi: 10.1088/0951-7715/26/12/3259.  Google Scholar

[31]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis,, Reprint of the 1984 edition, (1984).   Google Scholar

[32]

X. Wang and H. Wu, Long-time behavior for the Hele-Shaw-Cahn-Hilliard system,, \emph{Asymptot. Anal.}, 78 (2012), 217.   Google Scholar

[33]

X. Wang and Z. Zhang, Well-posedness of the Hele-Shaw-Cahn-Hilliard system,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 30 (2013), 367.  doi: 10.1016/j.anihpc.2012.06.003.  Google Scholar

[34]

S. M. Wise, Unconditionally stable finite difference, nonlinear multigrid simulation of the Cahn-Hilliard-Hele-Shaw system of equations,, \emph{J. Sci. Comput.}, 44 (2010), 38.  doi: 10.1007/s10915-010-9363-4.  Google Scholar

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