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

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).Keywords: Navier-Stokes equations, nonlocal Cahn-Hilliard equations, singular 1 potentials, incompressible binary fluids, global attractors, trajectory attractors.

Weak solutions for a diffuse interface model for two‐phase flows of incompressible fluids with different densities and nonlocal free energies

“…Finally, the estimates of F h ( ) and Ψ ′ 0 ( ) in L 2 (Ω) follow directly from (30) and (20). Now, we will prove existence of solution to the time-discrete system.…”

“…Concerning the nonlocal model H with a regular kernel, where the convective Cahn‐Hilliard equation is replaced by the convective nonlocal Cahn‐Hilliard equation with a regular kernel, first studies were done in references; see also Frigeri and the references there for more recent results. More recently, the nonlocal AGG model with a regular kernel, where the convective Cahn‐Hilliard equation is replaced by the convective nonlocal Cahn‐Hilliard equation with a regular kernel, was studied by Frigeri, and he showed the existence of a weak solution for that model.…”

“…12 -16 On the other hand, the nonlocal Cahn-Hilliard equation with a singular kernel was first analyzed in Abels et al, 17 where they proved the existence and uniqueness of a weak solution of the nonlocal Cahn-Hilliard equation, its regularity properties, and the existence of a (connected) global attractor. Concerning the nonlocal model H with a regular kernel, where the convective Cahn-Hilliard equation is replaced by the convective nonlocal Cahn-Hilliard equation with a regular kernel, first studies were done in references [18][19][20] ; see also Frigeri 21 and the references there for more recent results. More recently, the nonlocal AGG model with a regular kernel, where the convective Cahn-Hilliard equation is replaced by the convective nonlocal Cahn-Hilliard equation with a regular kernel, was studied by Frigeri, 22 and he showed the existence of a weak solution for that model.…”

On an Inviscid Model for Incompressible Two-Phase Flows with Nonlocal Interaction

“…This system assumes the case of matched densities for the two fluids and constant mobility. On the other hand, the system comprising of (1.6), (1.1), (1.3), subject to homogeneous Neumann and no slip boundary conditions for µ and u, respectively, has been analyzed recently in [8,9,10,12,13,11] under various assumptions on F, J and on the mobility and viscosity coefficients, respectively. We also recall that the nonlocal Cahn-Hilliard-Navier-Stokes system described earlier is a generalized version of the classical Cahn-Hilliard-Navier-Stokes system when in the place of aϕ − J * ϕ one usually finds −∆ϕ, see [1,2,5,7,14,15,16,26,27,28] and references therein.…”

Global well-posedness of strong solution for the three dimensional dynamic Cahn-Hilliard-Stokes model

“…Recently, the couplings of the Cahn-Hilliard equation with other basic modeling equations also have been proposed in various situation to describe complicated phenomena in fluid mechanics involving phase transition, such as the Cahn-Hilliard-Navier-Stokes (CHNS) equation [5,9,13,14,19], the Cahn-Hilliard-Hele-Shaw (CHHS) [4,12,[22][23][24] equation, and the Cahn-Hilliard-Boussinesq (CHB) [26] equation.…”