Abstract. We consider a general family of regularized models for incompressible two-phase flows based on the Allen-Cahn formulation in n-dimensional compact Riemannian manifolds for n = 2, 3. The system we consider consists of a regularized family of Navier-Stokes equations (including the Navier-Stokes-α-like model, the Leray-α model, the Modified Leray-α model, the Simplified Bardina model, the Navier-Stokes-Voight model and the Navier-Stokes model) for the fluid velocity u suitably coupled with a convective Allen-Cahn equation for the order (phase) parameter φ. We give a unified analysis of the entire three-parameter family of twophase models using only abstract mapping properties of the principal dissipation and smoothing operators, and then use assumptions about the specific form of the parameterizations, leading to specific models, only when necessary to obtain the sharpest results. We establish existence, stability and regularity results, and some results for singular perturbations, which as special cases include the inviscid limit of viscous models and the α → 0 limit in α-models. Then, we also show the existence of a global attractor and exponential attractor for our general model, and then establish precise conditions under which each trajectory (u, φ) converges to a single equilibrium by means of a Lojasiewicz-Simon inequality. We also derive new results on the existence of global and exponential attractors for the regularized family of Navier-Stokes equations and magnetohydrodynamics models which improve and complement the results of [44]. Finally, our analysis is applied to certain regularized Ericksen-Leslie (RSEL) models for the hydrodynamics of liquid crystals in n-dimensional compact Riemannian manifolds.

In this article, we consider a three‐dimensional Navier–Stokes–Voight model with memory where relaxation effects are described through a distributed delay. We prove the existence of uniform global attractors scriptAϵ, where ϵ ∈ (0,1) is the scaling parameter in the memory kernel. Furthermore, we prove that the model converges to the classical three‐dimensional Navier–Stokes–Voight system in an appropriate sense as ϵ → 0. In particular, we construct a family of exponential attractors Ξϵ that is robust as ϵ → 0. Copyright © 2013 John Wiley & Sons, Ltd.

In this article, we review from the mathematical and numerical points of view some of the recent progresses in the area of flow control.

The main objectives of this paper are to adapt an efficient and accurate spectralprojection method for a wind-driven, double-gyre, mid-latitude, quasi-geostrophic ocean model, and to study the double-gyre phenomenon from numerical and structural analysis points of view. A number of numerical simulations are carried out and their structural stability and structural transition/bifurcation are investigated using a new dynamical systems theory of two-dimensional incompressible flows.

We study in this article a class of robust control problems associated with a coupled Cahn-Hilliard-Navier-Stokes model in a two dimensional bounded domain. The model consists of the Navier-Stokes equations for the velocity, coupled with the Cahn-Hilliard model for the order (phase) parameter. We prove the existence and uniqueness of solutions and we derive a first-order necessary optimality condition for these robust control problems.2000 Mathematics Subject Classification. 93C05, 93B50, 93C35.