Nonlocal Cahn-Hilliard-Navier-Stokes systems with singular potentials

TL;DR

This paper studies a complex fluid mixture model combining nonlocal Cahn-Hilliard and Navier-Stokes equations with singular potentials, establishing existence of solutions and attractors in 2D and 3D, advancing understanding of physically relevant cases.

Contribution

It extends previous work to include singular potentials in nonlocal Cahn-Hilliard-Navier-Stokes systems, proving existence of solutions and attractors in both 2D and 3D cases.

Findings

Existence of global weak solutions in 2D with singular potentials

Existence of a global attractor for the 2D generalized semiflow

Existence of a trajectory attractor in 3D

Abstract

Here we consider a Cahn-Hilliard-Navier-Stokes system characterized by a nonlocal Cahn-Hilliard equation with a singular (e.g., logarithmic) potential. This system originates from a diffuse interface model for incompressible isothermal mixtures of two immiscible fluids. We have already analyzed the case of smooth potentials with arbitrary polynomial growth. Here, taking advantage of the previous results, we study this more challenging (and physically relevant) case. We first establish the existence of a global weak solution with no-slip and no-flux boundary conditions. Then we prove the existence of the global attractor for the 2D generalized semiflow (in the sense of J.M. Ball). We recall that uniqueness is still an open issue even in 2D. We also obtain, as byproduct, the existence of a connected global attractor for the (convective) nonlocal Cahn-Hilliard equation. Finally, in the 3D case, we establish the existence of a trajectory attractor (in the sense of V.V. Chepyzhov and M.I. Vishik).