Convergence to equilibrium of global weak solutions for a Cahn–Hilliard–Navier–Stokes vesicle model

Abstract

In this paper, we introduce a model describing the dynamic of vesicle membranes within an incompressible viscous fluid in 3D domains. The system consists of the Navier-Stokes equations, with an extra stress tensor depending on the membrane, coupled with a Cahn-Hilliard phase-field equation associated to a bending energy plus a penalization term related to the area conservation. This problem has a dissipative in time free-energy which leads, in particular, to prove the existence of global in time weak solutions. We analyze the large-time behavior of the weak solutions. By using a modified Lojasiewicz-Simon's result, we prove the convergence as time goes to infinity of each (whole) trajectory to a single equilibrium. Finally, the convergence of the trajectory of the phase is improved by imposing more regularity on the domain and initial phase.Numerous studies have been devoted to this type of models and a detailed description of them can be seen in [9] and references therein. A phase function is also used in subsequent papers to model vesicle membranes as diffuse interfaces. In [15] and [22], a coupled Allen-Cahn and Navier-Stokes problem is studied approaching both constrains, area and volume, via a penalization functional. On the other hand, without taking into account the vesiclefluid interaction, a Cahn-Hilliard phase-field model for vesicle membranes is introduced in [2], and the well-posedness of a phase-field approximation satisfying area and volume constraints pointwisely is established in [7].In this paper, a Cahn-Hilliard-Navier-Stokes model will be considered. Since the volume constraint is implicitly satisfied for the Cahn-Hilliard equation, only the surface area constraint will be approximated via penalization. Moreover, the resulting problem will be thermodynamically consistent because there exists a free-energy (kinetic plus bending plus penalized one) dissipative in time along the trajectories. This fact is used to prove the existence of global in time weak solutions.On the other hand, the large-time behavior of the solutions will be analyzed following the way of [12], [16], [4]. Firstly, we prove that the ω-limit set for weak solutions is composed by critical points of the free-energy. After that, by using a modified Lojasiewicz-Simon's result we demonstrate the convergence of the whole trajectory to a single equilibrium. Finally, the convergence is improved for the phase assuming more regular data, but without using strong regularity for the velocity and pressure variables.The main novelties in this paper are the following:• The introduction of a new Navier-Stokes-Cahn-Hilliard problem modeling vesicle-fluid interactions.• The proof of a modified Lojasiewicz-Simon's result associated to weak solutions of a fourth order elliptic problem, which is adapted to the exigences of this new model. In fact, it was not possible to apply any already known Lojasiewicz-Simon's result to this problem.• Given a global weak solution, we choose a special regularized energy satisfying the energy's law inequality eit...

Well-posedness of a Hydrodynamic Phase-field Model for Functionalized Membrane-Fluid Interaction

“…for almost all s ≥ 0 including s = 0 and all t ≥ s, by taking the available weak/strong subsequent convergence results into account and using the weak lower semicontinuity of norms. Since the compactness argument is standard (see, for instance, [5] for the Navier-Stokes-Cahn-Hilliard system, [14] for the functionalized Cahn-Hilliard equation and [11,17] for some different type of vesicle-fluid interaction models), we omit the details here. We proceed to derive some higher-order spatial estimates for ω and φ.…”

“…A lot of work has been done to understand morphological changes of membranes. We refer to [2,3,7,8,18-22, 33, 39] for mathematical modeling and numerical simulations, see also [9,11,13,17,23,37,57] for rigorous analysis.…”

“…However, we note that when both the volume and surface area constraints are imposed by Lagrange multipliers, analysis of the resulting hydrodynamic system (see [20]) remains open. On the other hand, the authors of [10,11] analyzed an alternative model, in which the phase-field function is governed by a convective Cahn-Hilliard type equation so that the volume conservation can be automatically guaranteed. Adopting the penalty of surface area constraint, the authors proved eventual regularity of global weak solutions and showed the convergence to equilibrium as times goes to infinity by using the Lojasiewicz-Simon approach.…”

“…In particular, the convergence to a single equilibrium (0, φ ∞ ) as t → +∞ for any bounded global weak/strong solutions, as well as Lyapunov stability of the zero velocity field and local (or global) minimizers of the elastic bending energy E(φ) (cf. [10,11,57] for related studies on some different type of vesicle-fluid interaction models). In this aspect, a suitable gradient inequality of Lojasiewicz-Simon type associated to the energy functional E(φ) will play a crucial role.…”

Longtime behavior of a second order finite element scheme simulating the kinematic effects in liquid crystal dynamics

“…Further, the time asymptotic behavior towards equilibrium points for a number of continuous and discrete models has been considered in the literature. In [6] the Cahn-Hilliard-Navier-Stokes vesicle model is considered. In the same area, one has the works of [1] where the authors studied the convergence to equilibrium for a second-order time semi-discretization.…”