Fractional Differential Equations

A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

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Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Roger Temam)

Nonlinear elliptic and parabolic equations of the second order

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.

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

Two different models for the evolution of incompressible binary fluid mixtures in a three-dimensional bounded domain are considered. They consist of the 3D incompressible Navier-Stokes equations, subject to time-dependent external forces and coupled with either a convective Allen-Cahn or Cahn-Hilliard equation. Such systems can be viewed as generalizations of the Navier-Stokes equations to two-phase fluids. Using the trajectory approach, the authors prove the existence of the trajectory attractor for both systems.

Trajectory attractors for binary fluid mixtures in 3D

We consider a model of nonisothermal phase transitions taking
place in a bounded spatial region. The order parameter $\psi$ is governed by
an Allen-Cahn type equation which is coupled with the equation for the
temperature $\theta$. The former is subject to a dynamic boundary condition
recently proposed by some physicists to account for interactions with the
walls. The latter is endowed with a boundary condition which can be a
standard one (Dirichlet, Neumann or Robin) or a dynamic one of Wentzell
type. We thus formulate a class of initial and boundary value problems whose
local existence and uniqueness is proven by means of a fixed point argument.
The local solution becomes global owing to suitable a priori estimates. Then
we analyze the asymptotic behavior of the solutions within the theory of
infinite-dimensional dynamical systems. In particular, we demonstrate the
existence of the global attractor as well as of an exponential attractor.

The non-isothermal Allen-Cahn equation with dynamic boundary conditions

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 nonlocal Cahn-Hilliard-Navier-Stokes systems in two dimensions

In this article, we propose a model of phase separation in a binary mixture confined to a bounded region which may be contained within porous walls. The boundary conditions are derived from a mass conservation law and variational methods. Employing classical methods, that is, fixed point theorems and standard energy methods, we obtain the existence and uniqueness of a global solution to our problem. We then also compare our model of phase separation with other previous Cahn–Hilliard equations with homogeneous Neumann and dynamic boundary conditions. Copyright © 2006 John Wiley & Sons, Ltd.

Ciprian G. Gal

Papers

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

Trajectory attractors for binary fluid mixtures in 3D

The non-isothermal Allen-Cahn equation with dynamic boundary conditions

On nonlocal Cahn-Hilliard-Navier-Stokes systems in two dimensions

A Cahn–Hilliard model in bounded domains with permeable walls