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 $\mathcal{A}$. Then we establish the existence of an exponential attractors $\mathcal{E}$. Thus $\mathcal{A}$ 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.

Keywords: Navier–Stokes equations, Incompressible fluids, Cahn–Hilliard equations, Two-phase flows, Global attractors, Exponential attractors, Fractal dimension, Convergence to equilibria

@article{AIHPC_2010__27_1_401_0, author = {Gal, Ciprian G. and Grasselli, Maurizio}, title = {Asymptotic behavior of a {Cahn{\textendash}Hilliard{\textendash}Navier{\textendash}Stokes} system in {2D}}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {401--436}, publisher = {Elsevier}, volume = {27}, number = {1}, year = {2010}, doi = {10.1016/j.anihpc.2009.11.013}, mrnumber = {2580516}, zbl = {1184.35055}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2009.11.013/} }

TY - JOUR AU - Gal, Ciprian G. AU - Grasselli, Maurizio TI - Asymptotic behavior of a Cahn–Hilliard–Navier–Stokes system in 2D JO - Annales de l'I.H.P. Analyse non linéaire PY - 2010 SP - 401 EP - 436 VL - 27 IS - 1 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.anihpc.2009.11.013/ DO - 10.1016/j.anihpc.2009.11.013 LA - en ID - AIHPC_2010__27_1_401_0 ER -

%0 Journal Article %A Gal, Ciprian G. %A Grasselli, Maurizio %T Asymptotic behavior of a Cahn–Hilliard–Navier–Stokes system in 2D %J Annales de l'I.H.P. Analyse non linéaire %D 2010 %P 401-436 %V 27 %N 1 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.anihpc.2009.11.013/ %R 10.1016/j.anihpc.2009.11.013 %G en %F 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, Volume 27 (2010) no. 1, pp. 401-436. doi : 10.1016/j.anihpc.2009.11.013. http://archive.numdam.org/articles/10.1016/j.anihpc.2009.11.013/

[1] 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

,[2] 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 | Zbl

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

, ,[4] 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 | EuDML | Numdam | MR | Zbl

, ,[5] Diffuse-interface methods in fluid mechanics, Annu. Rev. Fluid Mech. vol. 30, Annual Reviews, Palo Alto, CA (1998), 139-165 | MR

, , ,[6] Computation of multiphase systems with phase field models, J. Comput. Phys. 190 (2003), 371-397 | MR | Zbl

, , ,[7] Turbulence and coarsening in active and passive binary mixtures, Phys. Rev. Lett. 95 (2005), 224501

, , , ,[8] 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 | Zbl

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

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

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

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

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

,[14] Attractors representing turbulent flows, Mem. Amer. Math. Soc. 53 no. 314 (1985) | MR | Zbl

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

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

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

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

, ,[19] 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 | Zbl

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

, ,[21] 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 | Zbl

,[22] Exponential attractors for a nonlinear reaction–diffusion system in ${\mathbb{R}}^{3}$, C. R. Math. Acad. Sci. Paris 330 (2000), 713-718 | Zbl

, , ,[23] 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 | Zbl

,[24] Navier–Stokes Equations and Turbulence, Encyclopedia Math. Appl. vol. 83, Cambridge University Press, Cambridge (2001) | MR | Zbl

, , , ,[25] 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 | EuDML | Numdam | MR | Zbl

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

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

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

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

[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 | Zbl

[31] Generalizations of the Sobolev–Lieb–Thirring inequalities and applications to the dimension of attractors, Differential Integral Equations 1 (1998), 1-21 | MR | Zbl

, , ,[32] Uniform attractors for a phase-field model with memory and quadratic nonlinearity, Indiana Univ. Math. J. 48 (1999), 1395-1445 | MR | Zbl

, , ,[33] Asymptotic behavior of a nonisothermal viscous Cahn–Hilliard equation with inertial term, J. Differential Equations 239 (2007), 38-60 | MR | Zbl

, , ,[34] Two-phase binary fluids and immiscible fluids described by an order parameter, Math. Models Methods Appl. Sci. 6 (1996), 8-15 | MR | Zbl

, , ,[35] Systèmes dynamiques dissipatifs et applications, Masson, Paris (1991) | MR | Zbl

,[36] String phase in phase-separating fluids under shear flow, Phys. Rev. Lett. 74 (1995), 126-129

, , , ,[37] Theory of dynamical critical phenomena, Rev. Modern Phys. 49 (1977), 435-479

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

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

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

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

,[42] Finite element approximation of a Cahn–Hilliard–Navier–Stokes system, Interfaces Free Bound. 10 (2008), 15-43 | MR | Zbl

, , ,[43] On non-Newtonian incompressible fluids with phase transitions, Math. Methods Appl. Sci. 29 (2006), 1523-1541 | MR | Zbl

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

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

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

,[47] Mathematical Topics in Fluid Mechanics, vol. 1. Incompressible Models, Oxford Science Publications, Oxford (1996) | MR | Zbl

,[48] 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 | Zbl

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

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

,[51] 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 | Zbl

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

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

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

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

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