Global and Trajectory Attractors for a Nonlocal Cahn–Hilliard–Navier–Stokes System
 Sergio Frigeri,
 Maurizio Grasselli
 … show all 2 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
The Cahn–Hilliard–Navier–Stokes system is based on a wellknown 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 noslip and noflux 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.
 Abels, H. (2009) On a diffusive interface model for twophase flows of viscous, incompressible fluids with matched densities. Arch. Ration. Mech. Anal. 194: pp. 463506 CrossRef
 Abels, H. (2009) Existence of weak solutions for a diffuse interface model for viscous, incompressible fluids with general densities. Commun. Math. Phys. 289: pp. 4573 CrossRef
 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)
 Abels, H., Feireisl, E. (2008) On a diffuse interface model for a twophase flow of compressible viscous fluids. Indiana Univ. Math. J. 57: pp. 659698 CrossRef
 Anderson, D.M., McFadden, G.B., Wheeler, A.A. (1998) Diffuseinterface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30: pp. 139165 CrossRef
 Badalassi, V.E., Ceniceros, H., Banerjee, S. (2003) Computation of multiphase systems with phase field models. J. Comput. Phys. 190: pp. 371397 CrossRef
 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)
 Ball, J.M. (2004) Global attractors for damped semilinear wave equations. Discrete Contin. Dyn. Syst. 10: pp. 3152 CrossRef
 Bates, P.W., Han, J. (2005) The Neumann boundary problem for a nonlocal Cahn–Hilliard equation. J. Differ. Equ. 212: pp. 235277 CrossRef
 Bates, P.W., Han, J. (2005) The Dirichlet boundary problem for a nonlocal Cahn–Hilliard equation. J. Math. Anal. Appl. 311: pp. 289312 CrossRef
 Boyer, F. (1999) Mathematical study of multiphase flow under shear through order parameter formulation. Asymptot. Anal. 20: pp. 175212
 Boyer, F. (2001) Nonhomogeneous Cahn–Hilliard fluids. Ann. Inst. H. Poincaré Anal. Non Linéaire 18: pp. 225259 CrossRef
 Boyer, F. (2002) A theoretical and numerical model for the study of incompressible mixture flows. Comput. Fluids 31: pp. 4168 CrossRef
 Chepyzhov, V.V., Vishik, M.I.: Attractors for equations of mathematical physics, vol. 49. American Mathematical Society, Providence (2002)
 Chepyzhov, V.V., Vishik, M.I. (1997) Evolution equations and their trajectory attractors. J. Math. Pures Appl. 76: pp. 913964
 Cheskidov, A., Foias, C. (2006) On global attractors of the 3DNavier–Stokes equations. J. Differ. Equ. 231: pp. 714754 CrossRef
 Colli, P., Frigeri, S., Grasselli, M. (2012) Global existence of weak solutions to a nonlocal Cahn–Hilliard– Navier–Stokes system. J. Math. Anal. Appl. 386: pp. 428444 CrossRef
 Cutland, N.J. (2005) Global attractors for small samples and germs of 3D Navier–Stokes equations. Nonlinear Anal. 62: pp. 265281 CrossRef
 Doi, M.: Dynamics of domains and textures. In: McLeish, T.C. (ed.) Theoretical Challenges in the Dynamics of Complex Fluids , NATOASI Series, vol. 339, pp. 293–314. Kluwer Academic, Dordrecht (1997)
 Feng, X. (2006) Fully discrete finite element approximation of the Navier–Stokes–Cahn–Hilliard diffuse interface model for twophase flows. SIAM J. Num. Anal. 44: pp. 10491072 CrossRef
 Flandoli, F., Schmalfuss, B. (1999) Weak solutions and attractors for the 3dimensional Navier–Stokes equations with nonregular force. J. Dynam. Differ. Equ. 11: pp. 355398 CrossRef
 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)
 Gajewski, H. (2002) On a nonlocal model of nonisothermal phase separation. Adv. Math. Sci. Appl. 12: pp. 569586
 Gajewski, H., Zacharias, K. (2003) On a nonlocal phase separation model. J. Math. Anal. Appl. 286: pp. 1131 CrossRef
 Gal, C.G., Grasselli, M. (2010) Asymptotic behavior of a Cahn–Hilliard–Navier–Stokes system in 2D. Ann. Inst. H. Poincaré Anal. Non Linéaire 27: pp. 401436 CrossRef
 Gal, C.G., Grasselli, M. (2010) Trajectory attractors for binary fluid mixtures in 3D. Chinese Ann. Math. Ser. B 31: pp. 655678 CrossRef
 Gal, C.G., Grasselli, M. (2011) Instability of twophase flows: a lower bound on the dimension of the global attractor of the Cahn–Hilliard–Navier–Stokes system. Phys. D 240: pp. 629635 CrossRef
 Grasselli, M., Pražák, D. (2011) Longtime behavior of a diffuse interface model for binary fluid mixtures with shear dependent viscosity. Interfaces Free Bound. 13: pp. 507530 CrossRef
 Giacomin, G., Lebowitz, J.L. (1997) Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits. J. Stat. Phys. 87: pp. 3761 CrossRef
 Giacomin, G., Lebowitz, J.L. (1998) Phase segregation dynamics in particle systems with long range interactions. II. Phase motion. SIAM J. Appl. Math. 58: pp. 17071729 CrossRef
 Gurtin, M.E., Polignone, D., Viñals, J. (1996) Twophase binary fluids and immiscible fluids described by an order parameter. Math. Models Meth. Appl. Sci. 6: pp. 815 CrossRef
 Han, J. (2004) The Cauchy problem and steady state solutions for a nonlocal Cahn–Hilliard equation. Electron. J. Differ. Equ. 113: pp. 9
 Haspot, B. (2010) Existence of global weak solution for compressible fluid models with a capillary tensor for discontinuous interfaces. Differ. Integr. Equ. 23: pp. 899934
 Hohenberg, P.C., Halperin, B.I. (1977) Theory of dynamical critical phenomena. Rev. Mod. Phys. 49: pp. 435479 CrossRef
 Jasnow, D., Viñals, J. (1996) Coarsegrained description of thermocapillary flow. Phys. Fluids 8: pp. 660669 CrossRef
 Kapustyan, A.V., Valero, J. (2007) Weak and strong attractors for the 3D Navier–Stokes system. J. Differ. Equ. 240: pp. 249278 CrossRef
 Kay, D., Styles, V., Welford, R. (2008) Finite element approximation of a Cahn–Hilliard–Navier–Stokes system. Interfaces Free Bound. 10: pp. 543
 Kim, J., Kang, K., Lowengrub, J. (2004) Conservative multigrid methods for Cahn–Hilliard fluids. J. Comput. Phys. 193: pp. 511543 CrossRef
 Kloeden, P.E., Valero, J. (2010) The Kneser property of the weak solutions of the three dimensional Navier–Stokes equations. Discrete Contin. Dyn. Syst. 28: pp. 161179 CrossRef
 Liu, C., Shen, J. (2003) A phase field model for the mixture of two incompressible fluids and its approximation by a Fourierspectral method. Phys. D 179: pp. 211228 CrossRef
 Londen, S.O., Petzeltová, H. (2011) Convergence of solutions of a nonlocal phasefield system. Discrete Contin. Dyn. Syst. Ser. S 4: pp. 653670
 Lowengrub, J., Truskinovsky, L. (1998) Quasiincompressible Cahn–Hilliard fluids and topological transitions. Proc. R. Soc. Lond. A 454: pp. 26172654 CrossRef
 MarínRubio, P., Real, J. (2010) ullback attractors for 2DNavier–Stokes equations with delays in continuous and sublinear operators. Discrete Contin. Dyn. Syst. 26: pp. 9891006 CrossRef
 Morro, A. (2010) Phasefield models of Cahn–Hilliard Fluids and extra fluxes. Adv. Theor. Appl. Mech. 3: pp. 409424
 Rohde, C. (2005) On local and nonlocal Navier–StokesKorteweg systems for liquidvapour phase transitions. Z. Angew. Math. Mech. 85: pp. 839857 CrossRef
 Rohde, C.: A local and loworder Navier–StokesKorteweg 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)
 Rosa, R.M.S. (2006) Asymptotic regularity conditions for the strong convergence towards weak limit sets and weak attractors of the 3D Navier–Stokes equations. J. Differ. Equ. 229: pp. 257269 CrossRef
 Sell, G.R. (1996) Global attractors for the threedimensional Navier–Stokes equations. J. Dynam. Differ. Equ. 8: pp. 133 CrossRef
 Shen, J., Yang, X. (2010) Energy stable schemes for Cahn–Hilliard phasefield model of twophase incompressible flows. Chinese Ann. Math. Ser. B 31: pp. 743758 CrossRef
 Starovoitov, V.N. (1997) The dynamics of a twocomponent fluid in the presence of capillary forces. Math. Notes 62: pp. 244254 CrossRef
 Temam, R.: Navier–Stokes Equations and Nonlinear Functional Analysis, 2nd edn. In: CBMSNSF Regional Conference Series in Applied Mathematics, vol. 66. SIAM, Philadelphia (1995)
 Zhao, L., Wu, H., Huang, H. (2009) Convergence to equilibrium for a phasefield model for the mixture of two viscous incompressible fluids. Commun. Math. Sci. 7: pp. 939962
 Title
 Global and Trajectory Attractors for a Nonlocal Cahn–Hilliard–Navier–Stokes System
 Journal

Journal of Dynamics and Differential Equations
Volume 24, Issue 4 , pp 827856
 Cover Date
 20121201
 DOI
 10.1007/s1088401292723
 Print ISSN
 10407294
 Online ISSN
 15729222
 Publisher
 Springer US
 Additional Links
 Topics
 Keywords

 Navier–Stokes equations
 Nonlocal Cahn–Hilliard equations
 Incompressible binary fluids
 Global attractors
 Trajectory attractors
 35Q30
 37L30
 45K05
 76T99
 Authors

 Sergio Frigeri ^{(1)}
 Maurizio Grasselli ^{(2)}
 Author Affiliations

 1. Dipartimento di Matematica F. Enriques, Università degli Studi di Milano, Milano, 20133, Italy
 2. Dipartimento di Matematica F. Brioschi, Politecnico di Milano, Milano, 20133, Italy