Global and trajectory attractors for a nonlocal Cahn-Hilliard-Navier-Stokes system

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 an energy inequality. This allows us to establish the existence of the trajectory attractor also in dimension three with a time dependent external force.

35 pages; presented at INDI2011 in Gargnano

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• Asymptotic regularity conditions for the strong convergence towards weak limit sets and weak attractors of the 3D Navier-Stokes equations
• Existence of global weak solution for compressible fluid models with a capillary tensor for discontinuous interfaces

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• Pullback exponential attractors for a Cahn-Hilliard-Navier-Stokes system in 2D
• Existence and energy estimates of weak solutions for nonlocal Cahn--Hilliard equations on unbounded domains
• Long-time behavior of a nonlocal Cahn-Hilliard equation with reaction
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