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Attractors for the Navier-Stokes-Cahn-Hilliard system
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 |
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.
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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