Asymptotic behavior of a Cahn–Hilliard–Navier–Stokes system in 2D

Abstract

We consider a model for the flow of a mixture of two homogeneous and incompressible fluids in a two-dimensional bounded domain. The model consists of a Navier-Stokes equation governing the fluid velocity coupled with a convective Cahn-Hilliard equation for the relative density of atoms of one of the fluids. Endowing the system with suitable boundary and initial conditions, we analyze the asymptotic behavior of its solutions. First, we prove that the initial and boundary value problem generates a strongly continuous semigroup on a suitable phase-space which possesses the global attractor A. Then we establish the existence of an exponential attractors E. Thus A has finite fractal dimension. This dimension is then estimated from above in terms of the physical parameters. Moreover, assuming the potential to be real analytic and in absence of volume forces, we demonstrate that each trajectory converges to a single equilibrium. We also obtain a convergence rate estimate in the phase-space metric.

Computational Phase‐Field Modeling

“…This theory has been widely studied from the physical, the mathematical and the computational points of view Jacqmin, 1999;Gal and Grasselli, 2010;Kay et al, 2008;Boyer et al, 2010;Gal and Grasselli, 2011;Lowengrub and Truskinovsky, 1998;Colli et al, 2012;Kim et al, 2004;Liu and Shen, 2003). To derive the theory, let us assume that we have two immiscible components with volume fractions ϕ 1 and ϕ 2 , respectively.…”

A mathematical model for phase separation: A generalized Cahn-Hilliard equation

“…To account for fluid motion, some authors analyze the so-called Navier-Stokes-Cahn-Hilliard system (see for instance [17,20,25]). They substitute the partial derivative of c with respect to t with the material derivative in (6), i.e.…”

Convergence to equilibrium for a phase-field model for the mixture of two viscous incompressible fluids

“…A more complete mathematical theory of existence, uniqueness, regularity and asymptotic behavior of solutions to (1.1)-(1.3) with singular potential (1.8) as well as variable viscosity and constant mobility was given in [1]. In the recent paper [9], the authors considered system (1.1)-(1.6) in 2-D with constant viscosity and mobility. They proved existence of a global attractor as well as an exponential attractor and then showed the upper bound of fractal dimension of the global attractor (cf.…”

“…Under the basic assumption that the nonlinear term is analytic in the unknown function, the convergence of uniformly bounded global solutions to equilibria as time goes to infinity can be proven (cf. [2,5,8,9,17,18,24,25,32,33,34] and references therein). In particular, we refer to [24,32,34] for the convergence result concerning the single Cahn-Hilliard equation subject to various boundary conditions.…”