Strong solutions for two-dimensional nonlocal Cahn–Hilliard–Navier–Stokes systems

Abstract

A well-known diffuse interface model for incompressible isothermal mixtures of two immiscible fluids consists of the Navier-Stokes system coupled with a convective Cahn-Hilliard equation. In some recent contributions the standard Cahn-Hilliard equation has been replaced by its nonlocal version. The corresponding system is physically more relevant and mathematically more challenging. Indeed, the only known results are essentially the existence of a global weak solution and the existence of a suitable notion of global attractor for the corresponding dynamical system defined without uniqueness. In fact, even in the two-dimensional case, uniqueness of weak solutions is still an open problem. Here we take a step forward in the case of regular potentials. First we prove the existence of a (unique) strong solution in two dimensions. Then we show that any weak solution regularizes in finite time uniformly with respect to bounded sets of initial data. This result allows us to deduce that the global attractor is the union of all the bounded complete trajectories which are strong solutions. We also demonstrate that each trajectory converges to a single equilibrium, provided that the potential is real analytic and the external forces vanish.

Existence and energy estimates of weak solutions for nonlocal Cahn–Hilliard equations on an unbounded domain

“…Also, there are nonlocal Cahn-Hilliard-Navier-Stokes systems which model the evolution of an isothermal mixture of two incompressible fluids considering nonlocal interactions between the molecules. Nonlocal Cahn-Hilliard equations (see, for instance, [1,4,6,9,16,20,21,22]) and nonlocal Cahn-Hilliard-Navier-Stokes equations (see, for instance, [8,10,11,12,13,14]) have been studied by many authors. In particular, the nonlocal Cahn-Hilliard equation (see e.g., [1,4,16]) (E1) ∂ t ϕ − ∆µ = 0 in Ω × (0, T ), µ = a(·)ϕ − J * ϕ + G ′ (ϕ) in Ω × (0, T ) and the nonlocal Cahn-Hilliard-Navier-Stokes equation (see e.g., [8,11,12,14])…”

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

“…Recall that (3.27) holds. The proof of the first part of this lemma follows immediately from that of [12,Theorem 2] where a bound u ∈ L 2 (0, T ; W 2,2 (Ω) ∩ H) was used on the velocity. In fact, estimating in a more accurate way we can replace this bound in terms of that ω ∈ L ∞ (0, T ; L q (Ω)) for the vorticity for some q ∈ (2, ∞).…”

“…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.…”

“…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.…”

On a Diffuse Interface Model for Tumour Growth with Non-local Interactions and Degenerate Mobilities

“…The non-local Cahn-Hilliard equation has been studied intensively by many authors, see for example [2,3,26,27,28]. There has also been some focus towards coupling with fluid equations, such as Brinkman and Hele-Shaw flows [17] or Navier-Stokes flow [9,20,22,21,23,24]. For the non-local Cahn-Hilliard equation with source terms, analytic results such as well-posedness and long-time behavior have been obtained in [16,42] for prescribed source terms or Lipschitz source terms depending on the order parameter.…”