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


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.


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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© 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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