On Nonlocal Cahn–Hilliard–Navier–Stokes Systems in Two Dimensions

Abstract

We consider a diffuse interface model which describes the motion of an incompressible isothermal mixture of two immiscible fluids. This model consists of the Navier-Stokes equations coupled with a convective nonlocal Cahn-Hilliard equation.Several results were already proven by two of the present authors. However, in the two-dimensional case, the uniqueness of weak solutions was still open. Here we establish such a result even in the case of degenerate mobility and singular potential.Moreover, we show the weak-strong uniqueness in the case of viscosity depending on the order parameter, provided that either the mobility is constant and the potential is regular or the mobility is degenerate and the potential is singular. In the case of constant viscosity, on account of the uniqueness results we can deduce the connectedness of the global attractor whose existence was obtained in a previous paper. The uniqueness technique can be adapted to show the validity of a smoothing property for the difference of two trajectories which is crucial to establish the existence of an exponential attractor. The latter is established even in the case of variable viscosity, constant mobility and regular potential.

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

“…Our assumptions on F, J remain essentially the same as in [8,9,12,11,17], and actually we can require much less than there.…”

“…Therefore, it is not straightforward to extend the results of [7] to our system (1.1)-(1.5), especially in the light of recent results proven for the nonlocal Cahn-Hilliard-Navier-Stokes system. This becomes actually more interesting when the assumptions on the potential F and the interaction kernel J can remain the same as in the recent work of [11], where a complete theory was developed for the full Cahn-Hilliard-Navier-Stokes system with nonlocal interaction, constant mobility and variable viscosity. We also wish to point out that the results presented in this contribution also remain true in the absence of physical boundaries when Ω = R 2 or Ω ⊂ R 2 is a compact manifold without boundary (cf.…”

“…Indeed, in the case when u ≡ 0 in (1.1) the best one can hope for smooth solutions is that they are at most globally Holder continuous (see [17]) even when ϕ 0 ∈ C ∞ , F ∈ C ∞ and Ω is of class C ∞ provided that J is a symmetric kernel that belongs to W 1,1 loc (R 2 ) . The latter turns out to be also optimal [11,17]. In particular, it is not at all expected for ϕ to possess any higher order regularity in spaces like W k,p , k ≥ 3, unless J is smooth and non-singular.…”

“…In order to provide the final results we need to introduce an additional assumption on the kernel J exactly as in [11].…”

“…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.…”