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August  2022, 15(8): 2249-2274. doi: 10.3934/dcdss.2022118

Attractors for the Navier-Stokes-Cahn-Hilliard system

Andrea Giorgini 1,, and  Roger Temam 2,

1. 

Department of Mathematics, Imperial College London, London, SW7 2AZ, United Kingdom
2. 

Institute for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, IN 47405, USA

* Corresponding author: Andrea Giorgini

Dedicated to Professor Maurizio Grasselli on the occasion of his 60th birthday with friendship and best wishes

Received  February 2022 Published  August 2022 Early access  May 2022

We investigate the longtime behavior of the solutions to the Navier-Stokes-Cahn-Hilliard system (also known as Model H) with singular (e.g. Flory-Huggins) potential and non-constant viscosity. We prove that the initial and boundary value problem generates a strongly continuous semigroup on a suitable phase-space. Next, we establish the existence of the global attractor and of exponential attractors, and their regularity properties.

Keywords: Model H, Navier-Stokes-Cahn-Hilliard system, logarithmic potentials, attractors.
Mathematics Subject Classification: 35B40, 35Q35, 35K61, 37L30, 76T06.
Citation: Andrea Giorgini, Roger Temam. Attractors for the Navier-Stokes-Cahn-Hilliard system. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 2249-2274. doi: 10.3934/dcdss.2022118
References:
[1]

H. Abels, On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities, Arch. Ration. Mech. Anal., 194 (2009), 463-506.  doi: 10.1007/s00205-008-0160-2.

[2]

H. Abels, Longtime behavior of solutions of a Navier-Stokes/Cahn-Hilliard system, Proceedings of the Conference "Nonlocal and Abstract Parabolic Equations and Their Applications", Bedlewo, in: Banach Center Publ., Polish Acad. Sci., 2009, pp. 9–19. doi: 10.4064/bc86-0-1.

[3]

D. M. Anderson, G. B. McFadden and A. A. Wheeler, Diffuse-interface methods in fluid mechanics, Annu. Rev. Fluid Mech., 30 (1998), 139-165.  doi: 10.1146/annurev.fluid.30.1.139.

[4]

S. Berti, G. Boffetta, M. Cencini and A. Vulpiani, Turbulence and coarsening in active and passive binary mixtures, Phys. Rev. Lett., 95 (2005), 224501. 

[5]

S. Bosia and S. Gatti, Pullback exponential attractor for a Cahn-Hilliard-Navier-Stokes system in 2D, Dyn. Partial Differ. Equ., 11 (2014), 1-38.  doi: 10.4310/DPDE.2014.v11.n1.a1.

[6]

S. Bosia, M. Grasselli and A. Miranville, On the longtime behavior of a 2D hydrodynamic model for chemically reacting binary fluid mixtures, Math. Methods Appl. Sci., 37 (2014), 726-743.  doi: 10.1002/mma.2832.

[7]

F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation, Asympt. Anal., 20 (1999), 175-212. 

[8]

R. Chella and J. Vinals, Mixing of two-phase fluids by a cavity flow, Phys. Rev. E, 53 (1996), 3832-3840. 

[9]

A. Cheskidov and C. Foias, On global attractors of the 3D Navier-Stokes equations, J. Differential Equations, 231 (2006), 714-754.  doi: 10.1016/j.jde.2006.08.021.

[10]

C. G. Gal and M. Grasselli, Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes in $2D$, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 27 (2010), 401-436.  doi: 10.1016/j.anihpc.2009.11.013.

[11]

C. G. Gal and M. Grasselli, Instability of two-phase flows: A lower bound on the dimension of the global attractor of the Cahn-Hilliard-Navier-Stokes system, Phys. D, 240 (2011), 629-635.  doi: 10.1016/j.physd.2010.11.014.

[12]

A. Giorgini, M. Grasselli and A. Miranville, The Cahn-Hilliard-Oono equation with singular potential, Math. Models Meth. Appl. Sci., 27 (2017), 2485-2510.  doi: 10.1142/S0218202517500506.

[13]

A. Giorgini, A. Miranville and R. Temam, Uniqueness and regularity for the Navier-Stokes-Cahn-Hilliard system, SIAM J. Math. Anal., 51 (2019), 2535-2574.  doi: 10.1137/18M1223459.

[14]

M. E. Gurtin, D. Polignone and J. Viñals, Two-phase binary fluids and immiscible fluids described by an order parameter, Math. Models Methods Appl. Sci., 6 (1996), 815-831.  doi: 10.1142/S0218202596000341.

[15]

J. He and H. Wu, Global well-posedness of a Navier-Stokes-Cahn-Hilliard system with chemotaxis and singular potential in $2D$, J. Differential Equations, 297 (2021), 47-80.  doi: 10.1016/j.jde.2021.06.022.

[16]

P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Rev. Mod. Phys., 49 (1977), 435-479. 

[17]

D. Jacqmin, Calculation of two phase Navier-Stokes flows using phase-field modeling, J. Comput. Phys., 155 (1999), 96-127.  doi: 10.1006/jcph.1999.6332.

[18]

D. Jasnow and J. Vinãls, Coarse-grained description of thermo-capillary flow, Phys. Fluids, 8 (1996), 660-669. 

[19]

C. Liu and J. Shen, A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method, Phys. D, 179 (2003), 211-228.  doi: 10.1016/S0167-2789(03)00030-7.

[20]

A. Miranville, The Cahn-Hilliard Equation: Recent Advances and Applications, CBMS-NSF Regional Conf. Ser. in Appl. Math. 95, SIAM, Philadelphia, PA., 2019. doi: 10.1137/1.9781611975925.

[21]

A. Miranville and R. Temam, On the Cahn-Hilliard-Oono-Navier-Stokes equations wih singular potentials, Appl. Anal., 95 (2016), 2609-2624.  doi: 10.1080/00036811.2015.1102893.

[22]

A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials, Math. Meth. Appl. Sci., 27 (2004), 545-582.  doi: 10.1002/mma.464.

[23]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, Handbook of Differential Equations: Evolutionary Equations, IV, 103–200, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008. doi: 10.1016/S1874-5717(08)00003-0.

[24]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[25]

R. Temam, Navier-Stokes Equations, AMS Chelsea Publishing, Providence, 2001. doi: 10.1090/chel/343.

[26]

L. Zhao, H. Wu and H. Huang, Convergence to equilibrium for a phase-field model for the mixture of two incompressible fluids, Commun. Math. Sci., 7 (2009), 939-962.  doi: 10.4310/CMS.2009.v7.n4.a7.

show all references

References:
[1]

H. Abels, On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities, Arch. Ration. Mech. Anal., 194 (2009), 463-506.  doi: 10.1007/s00205-008-0160-2.

[2]

H. Abels, Longtime behavior of solutions of a Navier-Stokes/Cahn-Hilliard system, Proceedings of the Conference "Nonlocal and Abstract Parabolic Equations and Their Applications", Bedlewo, in: Banach Center Publ., Polish Acad. Sci., 2009, pp. 9–19. doi: 10.4064/bc86-0-1.

[3]

D. M. Anderson, G. B. McFadden and A. A. Wheeler, Diffuse-interface methods in fluid mechanics, Annu. Rev. Fluid Mech., 30 (1998), 139-165.  doi: 10.1146/annurev.fluid.30.1.139.

[4]

S. Berti, G. Boffetta, M. Cencini and A. Vulpiani, Turbulence and coarsening in active and passive binary mixtures, Phys. Rev. Lett., 95 (2005), 224501. 

[5]

S. Bosia and S. Gatti, Pullback exponential attractor for a Cahn-Hilliard-Navier-Stokes system in 2D, Dyn. Partial Differ. Equ., 11 (2014), 1-38.  doi: 10.4310/DPDE.2014.v11.n1.a1.

[6]

S. Bosia, M. Grasselli and A. Miranville, On the longtime behavior of a 2D hydrodynamic model for chemically reacting binary fluid mixtures, Math. Methods Appl. Sci., 37 (2014), 726-743.  doi: 10.1002/mma.2832.

[7]

F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation, Asympt. Anal., 20 (1999), 175-212. 

[8]

R. Chella and J. Vinals, Mixing of two-phase fluids by a cavity flow, Phys. Rev. E, 53 (1996), 3832-3840. 

[9]

A. Cheskidov and C. Foias, On global attractors of the 3D Navier-Stokes equations, J. Differential Equations, 231 (2006), 714-754.  doi: 10.1016/j.jde.2006.08.021.

[10]

C. G. Gal and M. Grasselli, Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes in $2D$, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 27 (2010), 401-436.  doi: 10.1016/j.anihpc.2009.11.013.

[11]

C. G. Gal and M. Grasselli, Instability of two-phase flows: A lower bound on the dimension of the global attractor of the Cahn-Hilliard-Navier-Stokes system, Phys. D, 240 (2011), 629-635.  doi: 10.1016/j.physd.2010.11.014.

[12]

A. Giorgini, M. Grasselli and A. Miranville, The Cahn-Hilliard-Oono equation with singular potential, Math. Models Meth. Appl. Sci., 27 (2017), 2485-2510.  doi: 10.1142/S0218202517500506.

[13]

A. Giorgini, A. Miranville and R. Temam, Uniqueness and regularity for the Navier-Stokes-Cahn-Hilliard system, SIAM J. Math. Anal., 51 (2019), 2535-2574.  doi: 10.1137/18M1223459.

[14]

M. E. Gurtin, D. Polignone and J. Viñals, Two-phase binary fluids and immiscible fluids described by an order parameter, Math. Models Methods Appl. Sci., 6 (1996), 815-831.  doi: 10.1142/S0218202596000341.

[15]

J. He and H. Wu, Global well-posedness of a Navier-Stokes-Cahn-Hilliard system with chemotaxis and singular potential in $2D$, J. Differential Equations, 297 (2021), 47-80.  doi: 10.1016/j.jde.2021.06.022.

[16]

P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Rev. Mod. Phys., 49 (1977), 435-479. 

[17]

D. Jacqmin, Calculation of two phase Navier-Stokes flows using phase-field modeling, J. Comput. Phys., 155 (1999), 96-127.  doi: 10.1006/jcph.1999.6332.

[18]

D. Jasnow and J. Vinãls, Coarse-grained description of thermo-capillary flow, Phys. Fluids, 8 (1996), 660-669. 

[19]

C. Liu and J. Shen, A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method, Phys. D, 179 (2003), 211-228.  doi: 10.1016/S0167-2789(03)00030-7.

[20]

A. Miranville, The Cahn-Hilliard Equation: Recent Advances and Applications, CBMS-NSF Regional Conf. Ser. in Appl. Math. 95, SIAM, Philadelphia, PA., 2019. doi: 10.1137/1.9781611975925.

[21]

A. Miranville and R. Temam, On the Cahn-Hilliard-Oono-Navier-Stokes equations wih singular potentials, Appl. Anal., 95 (2016), 2609-2624.  doi: 10.1080/00036811.2015.1102893.

[22]

A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials, Math. Meth. Appl. Sci., 27 (2004), 545-582.  doi: 10.1002/mma.464.

[23]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, Handbook of Differential Equations: Evolutionary Equations, IV, 103–200, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008. doi: 10.1016/S1874-5717(08)00003-0.

[24]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[25]

R. Temam, Navier-Stokes Equations, AMS Chelsea Publishing, Providence, 2001. doi: 10.1090/chel/343.

[26]

L. Zhao, H. Wu and H. Huang, Convergence to equilibrium for a phase-field model for the mixture of two incompressible fluids, Commun. Math. Sci., 7 (2009), 939-962.  doi: 10.4310/CMS.2009.v7.n4.a7.

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