Asymptotic behavior of a Cahn–Hilliard–Navier–Stokes system in 2D
Gal, Ciprian G. ; Grasselli, Maurizio
Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010), p. 401-436 / Harvested from Numdam
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We consider a model for the flow of a mixture of two homogeneous and incompressible fluids in a two-dimensional bounded domain. The model consists of a Navier–Stokes equation governing the fluid velocity coupled with a convective Cahn–Hilliard equation for the relative density of atoms of one of the fluids. Endowing the system with suitable boundary and initial conditions, we analyze the asymptotic behavior of its solutions. First, we prove that the initial and boundary value problem generates a strongly continuous semigroup on a suitable phase-space which possesses the global attractor 𝒜. Then we establish the existence of an exponential attractors ℰ. Thus 𝒜 has finite fractal dimension. This dimension is then estimated from above in terms of the physical parameters. Moreover, assuming the potential to be real analytic and in absence of volume forces, we demonstrate that each trajectory converges to a single equilibrium. We also obtain a convergence rate estimate in the phase-space metric.

Publié le : 2010-01-01
DOI : https://doi.org/10.1016/j.anihpc.2009.11.013
Classification:  35B40,  35B41,  35K55,  35Q35,  37L30,  76D05,  76T99 
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     author = {Gal, Ciprian G. and Grasselli, Maurizio},
     title = {Asymptotic behavior of a Cahn--Hilliard--Navier--Stokes system in 2D},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {27},
     year = {2010},
     pages = {401-436},
     doi = {10.1016/j.anihpc.2009.11.013},
     mrnumber = {2580516},
     zbl = {1184.35055},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2010__27_1_401_0}
}

Gal, Ciprian G.; Grasselli, Maurizio. Asymptotic behavior of a Cahn–Hilliard–Navier–Stokes system in 2D. Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) pp. 401-436. doi : 10.1016/j.anihpc.2009.11.013. http://gdmltest.u-ga.fr/item/AIHPC_2010__27_1_401_0/

[1] 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, Banach Center Publ. vol. 86, Polish Acad. Sci. (2009), 9-19 | Zbl 1167.76008

[2] 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 | MR 2563636 | Zbl 1254.76158

[3] H. Abels, E. Feireisl, On a diffuse interface model for a two-phase flow of compressible viscous fluids, Indiana Univ. Math. J. 57 (2008), 659-698 | MR 2414331 | Zbl 1144.35041

[4] H. Abels, M. Röger, Existence of weak solutions for a non-classical sharp interface model for a two-phase flow of viscous, incompressible fluids, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 no. 6 (2009), 2403-2424 | Numdam | MR 2569901 | Zbl 1181.35343

[5] D.M. Anderson, G.B. Mcfadden, A.A. Wheeler, Diffuse-interface methods in fluid mechanics, Annu. Rev. Fluid Mech. vol. 30, Annual Reviews, Palo Alto, CA (1998), 139-165 | MR 1609626

[6] V.E. Badalassi, H.D. Ceniceros, S. Banerjee, Computation of multiphase systems with phase field models, J. Comput. Phys. 190 (2003), 371-397 | MR 2013023 | Zbl 1076.76517

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

[8] M.A. Blinchevskaya, Yu.S. Ilyashenko, Estimate for the entropy dimension of the maximal attractor for k-contracting systems in an infinite-dimensional space, Russ. J. Math. Phys. 6 (1999), 20-26 | MR 1816785 | Zbl 1059.37507

[9] F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation, Asymptot. Anal. 20 (1999), 175-212 | MR 1700669 | Zbl 0937.35123

[10] F. Boyer, Nonhomogenous Cahn–Hilliard fluids, Ann. Inst. H. Poincaré Anal. Non Linéaire 18 (2001), 225-259 | Numdam | MR 1808030 | Zbl 1037.76062

[11] F. Boyer, A theoretical and numerical model for the study of incompressible model flows, Comput. & Fluids 31 (2002), 41-68 | Zbl 1057.76060

[12] F. Boyer, P. Fabrie, Persistency of 2D perturbations of one-dimensional solutions for a Cahn–Hilliard flow model under high shear, Asymptot. Anal. 33 (2003), 107-151 | MR 1977766 | Zbl 1137.76823

[13] A.J. Bray, Theory of phase-ordering kinetics, Adv. Phys. 51 (2002), 481-587

[14] P. Constantin, C. Foias, R. Temam, Attractors representing turbulent flows, Mem. Amer. Math. Soc. 53 no. 314 (1985) | MR 776345 | Zbl 0567.35070

[15] J.W. Cahn, On spinodal decomposition, Acta Metall. Mater. 9 (1961), 795-801

[16] J.W. Cahn, J.E. Hilliard, Free energy of a nonuniform system, I, interfacial free energy, J. Chem. Phys. 28 (1958), 258-267

[17] R. Chella, J. Viñals, Mixing of a two-phase fluid by a cavity flow, Phys. Rev. E 53 (1996), 3832-3840

[18] V.V. Chepyzhov, A.A. Ilyin, A note on the fractal dimension of attractors of dissipative dynamical systems, Nonlinear Anal. 44 (2001), 811-819 | MR 1825783 | Zbl 1153.37438

[19] V.V. Chepyzhov, A.A. Ilyin, On the fractal dimension of invariant sets: Applications to Navier–Stokes equations, Partial Differential Equations and Applications Discrete Contin. Dyn. Syst. 10 (2004), 117-135 | MR 2026186 | Zbl 1049.37047

[20] V.V. Chepyzhov, M.I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc. Colloq. Publ. vol. 49, American Mathematical Society, Providence, RI (2002) | MR 1868930 | Zbl 0986.35001

[21] L. Chupin, Existence result for a mixture of non Newtonian flows with stress diffusion using the Cahn–Hilliard formulation, Discrete Contin. Dyn. Syst. Ser. B 3 (2003), 45-68 | MR 1951567 | Zbl 1139.76304

[22] M. Efendiev, A. Miranville, S. Zelik, Exponential attractors for a nonlinear reaction–diffusion system in ℝ 3 , C. R. Math. Acad. Sci. Paris 330 (2000), 713-718 | Zbl 1151.35315

[23] X. Feng, Fully discrete finite element approximation of the Navier–Stokes–Cahn–Hilliard diffuse interface model for two-phase flows, SIAM J. Numer. Anal. 44 (2006), 1049-1072 | MR 2231855 | Zbl 1344.76052

[24] C. Foias, O. Manley, R. Rosa, R. Temam, Navier–Stokes Equations and Turbulence, Encyclopedia Math. Appl. vol. 83, Cambridge University Press, Cambridge (2001) | MR 1855030 | Zbl 0994.35002

[25] C. Foias, G. Prodi, Sur le comportement global des solutions non stationnaires des equations de Navier–Stokes en dimension deux, Rend. Semin. Mat. Univ. Padova 39 (1967), 1-34 | Numdam | MR 223716 | Zbl 0176.54103

[26] C. Foias, R. Temam, Some analytic and geometric properties of the solution of the Navier–Stokes equations, J. Math. Pures Appl. (9) 58 (1979), 339-368 | MR 544257 | Zbl 0454.35073

[27] H. Gajewski, A.-J. Griepentrog, A descent method for the free energy of multicomponent systems, Discrete Contin. Dyn. Syst. 15 (2006), 505-528 | MR 2199441 | Zbl 1141.90500

[28] C.G. Gal, M. Grasselli, Longtime behavior of a model for homogeneous incompressible two-phase flows, submitted for publication | MR 2629471

[29] C.G. Gal, M. Grasselli, Trajectory attractors for binary fluid mixtures in 3D, submitted for publication | MR 2726061 | Zbl 1223.35079

[30] C.G. Gal, M. Grasselli, Instability of two-phase flows: A lower bound on the dimension of the global attractor of the Cahn–Hilliard–Navier–Stokes system, submitted for publication | MR 2770979 | Zbl 1214.37055

[31] J.M. Ghidaglia, M. Marion, R. Temam, Generalizations of the Sobolev–Lieb–Thirring inequalities and applications to the dimension of attractors, Differential Integral Equations 1 (1998), 1-21 | MR 920485 | Zbl 0745.46037

[32] C. Giorgi, M. Grasselli, V. Pata, Uniform attractors for a phase-field model with memory and quadratic nonlinearity, Indiana Univ. Math. J. 48 (1999), 1395-1445 | MR 1757078 | Zbl 0940.35037

[33] M. Grasselli, H. Petzeltová, G. Schimperna, Asymptotic behavior of a nonisothermal viscous Cahn–Hilliard equation with inertial term, J. Differential Equations 239 (2007), 38-60 | MR 2341548 | Zbl 1129.35017

[34] M.E. Gurtin, D. Polignone, J. Viñals, Two-phase binary fluids and immiscible fluids described by an order parameter, Math. Models Methods Appl. Sci. 6 (1996), 8-15 | MR 1404829 | Zbl 0857.76008

[35] A. Haraux, Systèmes dynamiques dissipatifs et applications, Masson, Paris (1991) | MR 1084372 | Zbl 0726.58001

[36] T. Hashimoto, K. Matsuzaka, E. Moses, A. Onuki, String phase in phase-separating fluids under shear flow, Phys. Rev. Lett. 74 (1995), 126-129

[37] P.C. Hohenberg, B.I. Halperin, Theory of dynamical critical phenomena, Rev. Modern Phys. 49 (1977), 435-479

[38] A.A. Ilyin, Lieb–Thirring integral inequalities and their applications to the attractors of the Navier–Stokes equations, Sb. Math. 196 (2005), 29-61 | Zbl 1083.35093

[39] D. Jacqmin, Calculation of two-phase Navier–Stokes flows using phase-field modelling, J. Comput. Phys. 155 (1999), 96-127 | MR 1716497 | Zbl 0966.76060

[40] D. Jasnow, J. Viñals, Coarse-grained description of thermo-capillary flow, Phys. Fluids 8 (1996), 660-669 | Zbl 1025.76521

[41] M.A. Jendoubi, A simple unified approach to some convergence theorem of L. Simon, J. Funct. Anal. 153 (1998), 187-202 | MR 1609269 | Zbl 0895.35012

[42] D. Kay, V. Styles, R. Welford, Finite element approximation of a Cahn–Hilliard–Navier–Stokes system, Interfaces Free Bound. 10 (2008), 15-43 | MR 2383535 | Zbl 1144.35043

[43] N. Kim, L. Consiglieri, J.F. Rodrigues, On non-Newtonian incompressible fluids with phase transitions, Math. Methods Appl. Sci. 29 (2006), 1523-1541 | MR 2249576 | Zbl 1101.76004

[44] J. Kim, K. Kang, J. Lowengrub, Conservative multigrid methods for Cahn–Hilliard fluids, J. Comput. Phys. 193 (2004), 511-543 | MR 2030475 | Zbl 1109.76348

[45] O.A. Ladyzhenskaya, A dynamical system generated by Navier–Stokes equations, J. Soviet Math. 3 (1975), 458-479 | Zbl 0336.35081

[46] F.-H. Lin, Nonlinear theory of defects in nematic liquid crystals: Phase transition and flow phenomena, Comm. Pure Appl. Math. 42 (1989), 789-814 | MR 1003435 | Zbl 0703.35173

[47] P.-L. Lions, Mathematical Topics in Fluid Mechanics, vol. 1. Incompressible Models, Oxford Science Publications, Oxford (1996) | MR 1422251 | Zbl 0866.76002

[48] C. Liu, 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 | MR 1984386 | Zbl 1092.76069

[49] J. Lowengrub, L. Truskinovsky, Quasi-incompressible Cahn–Hilliard fluids and topological transitions, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 454 (1998), 2617-2654 | MR 1650795 | Zbl 0927.76007

[50] A. Onuki, Phase transitions of fluids in shear flow, J. Phys.: Condens. Matter 9 (1997), 6119-6157

[51] J.C. Robinson, Infinite-Dimensional Dynamical Systems. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge Texts Appl. Math., Cambridge University Press, Cambridge (2001) | MR 1881888 | Zbl 0980.35001

[52] R. Ruiz, D.R. Nelson, Turbulence in binary fluid mixtures, Phys. Rev. A 23 (1981), 3224-3246

[53] E.D. Siggia, Late stages of spinodal decomposition in binary mixtures, Phys. Rev. A 20 (1979), 595-605

[54] V.N. Starovoitov, The dynamics of a two-component fluid in the presence of capillary forces, Math. Notes 62 (1997), 244-254 | MR 1619861 | Zbl 0921.35134

[55] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Appl. Math. Sci. vol. 68, Springer-Verlag, New York (1997) | MR 1441312 | Zbl 0871.35001