Attractors for the Navier-Stokes-Cahn-Hilliard system

Andrea Giorgini; Roger Temam

doi:10.3934/dcdss.2022118

Article Contents

2022, Volume 15, Issue 8: 2249-2274. Doi: 10.3934/dcdss.2022118

This issue Previous Article An oxygen driven proliferative-to-invasive transition of glioma cells: An analytical study Next Article Trajectory attractors for 3D damped Euler equations and their approximation

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
^* 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

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: 35B40, 35Q35, 35K61, 37L30, 76T06.

Citation:

Full Text(HTML)

Related Papers

Cited by

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.