Dynamics of Partial Differential Equations

Volume 9 (2012)

Number 4

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

Pages: 273 – 304

DOI: http://dx.doi.org/10.4310/DPDE.2012.v9.n4.a1


Sergio Frigeri (Dipartimento di Matematica F. Enriques, Università degli Studi di Milano, Italy)

Maurizio Grasselli (Dipartimento di Matematica F. Brioschi, Politecnico di Milano, Italy)


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).


Navier-Stokes equations, nonlocal Cahn-Hilliard equations, singular potentials, incompressible binary fluids, global attractors, trajectory attractors

2010 Mathematics Subject Classification

35Q30, 37L30, 45K05, 76Txx

Full Text (PDF format)