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

TL;DR

This paper studies the long-term behavior of a nonlocal Cahn-Hilliard-Navier-Stokes system, proving the existence of global and trajectory attractors in various dimensions using energy methods and generalized semiflows.

Contribution

It establishes the existence of global and trajectory attractors for a nonlocal coupled fluid system, extending previous results to three dimensions and time-dependent forces.

Findings

Existence of a global attractor in two dimensions.

Existence of a connected global attractor in three dimensions.

Trajectory attractor existence under energy inequality in three dimensions.

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.