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Journal of Dynamics and Differential Equations

, Volume 24, Issue 4, pp 827–856 | Cite as

Global and Trajectory Attractors for a Nonlocal Cahn–Hilliard–Navier–Stokes System

  • Sergio Frigeri
  • Maurizio GrasselliEmail author
Article

Abstract

The Cahn–Hilliard–Navier–Stokes system is based on a well-known diffuse interface model and describes the evolution of an incompressible isothermal mixture of binary fluids. A nonlocal variant consists of the Navier–Stokes equations suitably coupled with a nonlocal Cahn–Hilliard equation. The authors, jointly with P. Colli, have already proven the existence of a global weak solution to a nonlocal Cahn–Hilliard–Navier–Stokes system subject to no-slip and no-flux boundary conditions. Uniqueness is still an open issue even in dimension two. However, in this case, the energy identity holds. This property is exploited here to define, following J.M. Ball’s approach, a generalized semiflow which has a global attractor. Through a similar argument, we can also show the existence of a (connected) global attractor for the convective nonlocal Cahn–Hilliard equation with a given velocity field, even in dimension three. Finally, we demonstrate that any weak solution fulfilling the energy inequality also satisfies a dissipative estimate. This allows us to establish the existence of the trajectory attractor also in dimension three with a time dependent external force.

Keywords

Navier–Stokes equations Nonlocal Cahn–Hilliard equations Incompressible binary fluids Global attractors Trajectory attractors 

Mathematics Subject Classification (2010)

35Q30 37L30 45K05 76T99 

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References

  1. 1.
    Abels H.: On a diffusive interface model for two-phase flows of viscous, incompressible fluids with matched densities. Arch. Ration. Mech. Anal. 194, 463–506 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Abels H.: Existence of weak solutions for a diffuse interface model for viscous, incompressible fluids with general densities. Commun. Math. Phys. 289, 45–73 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Abels, H.: Longtime behavior of solutions of a Navier–Stokes/Cahn–Hilliard system. In: Proceedings of the Conference Nonlocal and Abstract Parabolic Equations and their Applications, vol. 86, pp. 9–19. Banach Center, Bedlewo (2009)Google Scholar
  4. 4.
    Abels H., Feireisl E.: On a diffuse interface model for a two-phase flow of compressible viscous fluids. Indiana Univ. Math. J. 57, 659–698 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Anderson D.M., McFadden G.B., Wheeler A.A.: Diffuse-interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30, 139–165 (1998)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Badalassi V.E., Ceniceros H., Banerjee S.: Computation of multiphase systems with phase field models. J. Comput. Phys. 190, 371–397 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Ball, J.M.: Continuity properties and global attractors of generalized semiflows and the Navier–Stokes equation. J. Nonlinear Sci. 7 (1997), 475–502 ; Erratum, J. Nonlinear Sci. 8, 233 (1998)Google Scholar
  8. 8.
    Ball J.M.: Global attractors for damped semilinear wave equations. Discrete Contin. Dyn. Syst. 10, 31–52 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Bates P.W., Han J.: The Neumann boundary problem for a nonlocal Cahn–Hilliard equation. J. Differ. Equ. 212, 235–277 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Bates P.W., Han J.: The Dirichlet boundary problem for a nonlocal Cahn–Hilliard equation. J. Math. Anal. Appl. 311, 289–312 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Boyer F.: Mathematical study of multi-phase flow under shear through order parameter formulation. Asymptot. Anal. 20, 175–212 (1999)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Boyer F.: Nonhomogeneous Cahn–Hilliard fluids. Ann. Inst. H. Poincaré Anal. Non Linéaire 18, 225–259 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Boyer F.: A theoretical and numerical model for the study of incompressible mixture flows. Comput. Fluids 31, 41–68 (2002)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Chepyzhov, V.V., Vishik, M.I.: Attractors for equations of mathematical physics, vol. 49. American Mathematical Society, Providence (2002)Google Scholar
  15. 15.
    Chepyzhov V.V., Vishik M.I.: Evolution equations and their trajectory attractors. J. Math. Pures Appl. 76, 913–964 (1997)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Cheskidov A., Foias C.: On global attractors of the 3D-Navier–Stokes equations. J. Differ. Equ. 231, 714–754 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Colli P., Frigeri S., Grasselli M.: Global existence of weak solutions to a nonlocal Cahn–Hilliard– Navier–Stokes system. J. Math. Anal. Appl. 386, 428–444 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Cutland N.J.: Global attractors for small samples and germs of 3D Navier–Stokes equations. Nonlinear Anal. 62, 265–281 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Doi, M.: Dynamics of domains and textures. In: McLeish, T.C. (ed.) Theoretical Challenges in the Dynamics of Complex Fluids , NATO-ASI Series, vol. 339, pp. 293–314. Kluwer Academic, Dordrecht (1997)Google Scholar
  20. 20.
    Feng X.: Fully discrete finite element approximation of the Navier–Stokes–Cahn–Hilliard diffuse interface model for two-phase flows. SIAM J. Num. Anal. 44, 1049–1072 (2006)zbMATHCrossRefGoogle Scholar
  21. 21.
    Flandoli F., Schmalfuss B.: Weak solutions and attractors for the 3-dimensional Navier–Stokes equations with nonregular force. J. Dynam. Differ. Equ. 11, 355–398 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Foias, C., Temam, R.: The connection between the Navier–Stokes equations, dynamical systems, and turbulence theory. In: Proceedings of Directions in Partial Differential Equations, 1985, Mathematics Research Center and University, Madison, vol. 54, pp. 55–73. Academic Press, Boston (1987)Google Scholar
  23. 23.
    Gajewski H.: On a nonlocal model of non-isothermal phase separation. Adv. Math. Sci. Appl. 12, 569–586 (2002)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Gajewski H., Zacharias K.: On a nonlocal phase separation model. J. Math. Anal. Appl. 286, 11–31 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Gal C.G., Grasselli M.: Asymptotic behavior of a Cahn–Hilliard–Navier–Stokes system in 2D. Ann. Inst. H. Poincaré Anal. Non Linéaire 27, 401–436 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Gal C.G., Grasselli M.: Trajectory attractors for binary fluid mixtures in 3D. Chinese Ann. Math. Ser. B 31, 655–678 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Gal C.G., Grasselli M.: 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, 629–635 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Grasselli M., Pražák D.: Longtime behavior of a diffuse interface model for binary fluid mixtures with shear dependent viscosity. Interfaces Free Bound. 13, 507–530 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Giacomin G., Lebowitz J.L.: Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits. J. Stat. Phys. 87, 37–61 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Giacomin G., Lebowitz J.L.: Phase segregation dynamics in particle systems with long range interactions. II. Phase motion. SIAM J. Appl. Math. 58, 1707–1729 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Gurtin M.E., Polignone D., Viñals J.: Two-phase binary fluids and immiscible fluids described by an order parameter. Math. Models Meth. Appl. Sci. 6, 8–15 (1996)CrossRefGoogle Scholar
  32. 32.
    Han J.: The Cauchy problem and steady state solutions for a nonlocal Cahn–Hilliard equation. Electron. J. Differ. Equ. 113, 9 (2004)Google Scholar
  33. 33.
    Haspot B.: Existence of global weak solution for compressible fluid models with a capillary tensor for discontinuous interfaces. Differ. Integr. Equ. 23, 899–934 (2010)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Hohenberg P.C., Halperin B.I.: Theory of dynamical critical phenomena. Rev. Mod. Phys. 49, 435–479 (1977)CrossRefGoogle Scholar
  35. 35.
    Jasnow D., Viñals J.: Coarse-grained description of thermo-capillary flow. Phys. Fluids 8, 660–669 (1996)zbMATHCrossRefGoogle Scholar
  36. 36.
    Kapustyan A.V., Valero J.: Weak and strong attractors for the 3D Navier–Stokes system. J. Differ. Equ. 240, 249–278 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Kay D., Styles V., Welford R.: Finite element approximation of a Cahn–Hilliard–Navier–Stokes system. Interfaces Free Bound. 10, 5–43 (2008)MathSciNetGoogle Scholar
  38. 38.
    Kim J., Kang K., Lowengrub J.: Conservative multigrid methods for Cahn–Hilliard fluids. J. Comput. Phys. 193, 511–543 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Kloeden P.E., Valero J.: The Kneser property of the weak solutions of the three dimensional Navier–Stokes equations. Discrete Contin. Dyn. Syst. 28, 161–179 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Liu C., Shen J.: A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method. Phys. D 179, 211–228 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Londen S.-O., Petzeltová H.: Convergence of solutions of a non-local phase-field system. Discrete Contin. Dyn. Syst. Ser. S 4, 653–670 (2011)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Lowengrub J., Truskinovsky L.: Quasi-incompressible Cahn–Hilliard fluids and topological transitions. Proc. R. Soc. Lond. A 454, 2617–2654 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Marín-Rubio P., Real J.: ullback attractors for 2D-Navier–Stokes equations with delays in continuous and sub-linear operators. Discrete Contin. Dyn. Syst. 26, 989–1006 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Morro A.: Phase-field models of Cahn–Hilliard Fluids and extra fluxes. Adv. Theor. Appl. Mech. 3, 409–424 (2010)Google Scholar
  45. 45.
    Rohde C.: On local and non-local Navier–Stokes-Korteweg systems for liquid-vapour phase transitions. Z. Angew. Math. Mech. 85, 839–857 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Rohde, C.: A local and low-order Navier–Stokes-Korteweg system. In: Holden, H., Karlsen, K.H. (eds.) Nonlinear partial differential equations and hyperbolic wave phenomena, vol. 526, pp. 315–337. American Mathematical Society, Providence (2010)Google Scholar
  47. 47.
    Rosa R.M.S.: Asymptotic regularity conditions for the strong convergence towards weak limit sets and weak attractors of the 3D Navier–Stokes equations. J. Differ. Equ. 229, 257–269 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Sell G.R.: Global attractors for the three-dimensional Navier–Stokes equations. J. Dynam. Differ. Equ. 8, 1–33 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Shen J., Yang X.: Energy stable schemes for Cahn–Hilliard phase-field model of two-phase incompressible flows. Chinese Ann. Math. Ser. B 31, 743–758 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Starovoitov V.N.: The dynamics of a two-component fluid in the presence of capillary forces. Math. Notes 62, 244–254 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Temam, R.: Navier–Stokes Equations and Nonlinear Functional Analysis, 2nd edn. In: CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 66. SIAM, Philadelphia (1995)Google Scholar
  52. 52.
    Zhao L., Wu H., Huang H.: Convergence to equilibrium for a phase-field model for the mixture of two viscous incompressible fluids. Commun. Math. Sci. 7, 939–962 (2009)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Dipartimento di Matematica F. EnriquesUniversità degli Studi di MilanoMilanoItaly
  2. 2.Dipartimento di Matematica F. BrioschiPolitecnico di MilanoMilanoItaly

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