Global attractor for the Navier–Stokes equations with the heat convection in a cylindrical pipe

Global existence of regular solutions to the Navier–Stokes equations coupled with the heat convection in a cylindrical pipe has already been shown. In this paper, we prove the existence of the global attractor to the equations and convergence of their solutions to a stationary one. Copyright © 2012 John Wiley & Sons, Ltd.     

Ladder estimates for micropolar fluid equations and regularity of global attractor

This paper is devoted to obtain ladder inequalities for 2D micropolar fluid equations on a periodic domain Q=(0, L)². The ladder inequalities are differential inequalities that connect the evolution of L² norms of derivatives of order N with the evolution of the L² norms of derivatives of other (usually lower) order. Moreover, we find (with slight assumption on external fields) long‐time upper bounds on the L² norms of derivatives of every order, which implies that a global attractor is made up from C^∞ functions. Copyright © 2010 John Wiley & Sons, Ltd.     

Pullback attractor for heat convection problem in a micropolar fluid

We consider a two-dimensional micropolar fluid flow heated from below. We assume that the temperature of the lower part of the boundary is a function of time. That leads to the non-autonomous system of equations. We show the existence of the pullback attractor for the problem. Next, the dimension of the attractor is estimated from above.     

Long time behaviour of a flow in infinite pipe conforming to slip boundary conditions

The paper analyses long time behaviour of solutions of the Navier–Stokes equations in a two‐dimensional pipe‐like domain. The system is studied with perfect slip boundary conditions with arbitrary inflow conditions at infinity. The main results show the existence of global in time solutions and of an attractor for the dynamical system generated by the model. The paper also establishes an upper bound for the Hausdorff dimension of the attractor. Copyright © 2005 John Wiley & Sons, Ltd.     

Attractor dimension estimate for plane shear flow of micropolar fluid with free boundary

This research is motivated by a problem from lubrication theory. We consider a free boundary problem of a two‐dimensional boundary‐driven micropolar fluid flow. The existence of a unique global‐in‐time solution of the problem and the global attractor for the associated semigroup are known. In this paper we estimate the dimension of the global attractor in terms of the given data and the geometry of the domain of the flow by establishing a new version of the Lieb–Thirring inequality with constants depending explicitly on the geometry of the domain. We also obtain some new estimates for the Navier–Stokes shear flows. Copyright © 2005 John Wiley & Sons, Ltd.     

Global well‐posedness result for density‐dependent incompressible viscous fluid in with linearly growing initial velocity

In this paper, we consider the density‐dependent incompressible Navier–Stokes equations in with linearly growing initial velocity at infinity. We obtain a blow‐up criterion and global well‐posedness of the two‐dimensional system. It generalized the local well‐posedness results due to the recent work by the first and third authors to the global well‐posedness in . Copyright © 2012 John Wiley & Sons, Ltd.     

On existence and regularity of solutions for 2‐D micropolar fluid equations with periodic boundary conditions

In this paper, we prove the existence and uniqueness of a global solution for 2‐D micropolar fluid equation with periodic boundary conditions. Then we restrict ourselves to the autonomous case and show the existence of a global attractor. Copyright © 2006 John Wiley & Sons, Ltd.     

Dimension of the global attractor for damped nonlinear wave equations

An estimate on the Hausdorff dimension of the global attractor for damped nonlinear wave equations, in two cases of nonlinear damping and linear damping, with Dirichlet boundary condition is obtained. The gained Hausdorff dimension is bounded and is independent of the concrete form of nonlinear damping term. In the case of linear damping, the gained Hausdorff dimension remains small for large damping, which conforms to the physical intuition.

     

Pullback exponential attractors for the non-autonomous micropolar fluid flows

This paper investigates the pullback asymptotic behaviors for the non-autonomous micropolar fluid flows in 2D bounded domains. We use the energy method, combining with some important properties of the generated processes, to prove the existence of pullback exponential attractors and global pullback attractors and show that they both with finite fractal dimension. Further, we give the relationship between global pullback attractors and pullback exponential attractors.     

Global attractor for the semilinear thermoelastic problem

In this paper we consider a class of semilinear thermoelastic problems. The global attractor for this semilinear thermoelastic problem with Dirichlet boundary condition is obtained. Copyright © 2003 John Wiley & Sons, Ltd.     

Global large solutions to incompressible Navier‐Stokes equations with gravity

In this per, we consider a special class of initial data for the three‐dimensional incompressible Navier–Stokes equations with gravity. We show that, under such conditions, the incompressible Navier‐Stokes equations with gravity are globally well posed, and the velocity minus gravity term has finite energy. The important features of the initial data is that the velocity fields minus gravity term are almost parallel to the corresponding vorticity fields in a very large space domain. Copyright © 2014 John Wiley & Sons, Ltd.     

Pullback attractors of non-autonomous micropolar fluid flows

Pullback attractors of the two-dimensional non-autonomous micropolar fluid motion model in a bounded domain are investigated. It is shown that a compact pullback attractor in H¹³(Ω) exists when its external driven function is translation bounded with respect to L₂³(Ω).     

Existence of global strong solution to the micropolar fluid system in a bounded domain

In this paper we are concerned with the initial boundary value problem of the micropolar fluid system in a three dimensional bounded domain. We study the resolvent problem of the linearized equations and prove the generation of analytic semigroup and its time decay estimates. In particular, L^p–L^q type estimates are obtained. By use of the L^p–L^q estimates for the semigroup, we prove the existence theorem of global in time solution to the original nonlinear problem for small initial data. Furthermore, we study the magneto‐micropolar fluid system in the final section. Copyright © 2005 John Wiley & Sons, Ltd.     

A global attractor for a fluid–plate interaction model accounting only for longitudinal deformations of the plate

We study asymptotic dynamics of a coupled system consisting of linearized 3D Navier–Stokes equations in a bounded domain and the classical (nonlinear) elastic plate equation for in‐plane motions on a flexible flat part of the boundary. The main novelty of the model is the assumption that the transversal displacements of the plate are negligible relative to in‐plane displacements. These kinds of models arise in the study of blood flows in large arteries. Our main result states the existence of a compact global attractor of finite dimension. Under some conditions this attractor is an exponentially attracting single point. We also show that the corresponding linearized system generates an exponentially stable C₀‐semigroup. We do not assume any kind of mechanical damping in the plate component. Thus our results mean that dissipation of the energy in the fluid because of viscosity is sufficient to stabilize the system. Copyright © 2011 John Wiley & Sons, Ltd.     

Global weak solutions to a ferrofluid flow model

We are concerned with the global solvability of the differential system introduced by Shliomis to describe the flow of a colloidal suspension of magnetized nanoparticles in a nonconducting liquid, under the action of an external magnetic field. The system is a combination of the Navier–Stokes equations, the magnetization equation, and the magnetostatic equations. We prove, by using a method of regularization, the existence of global‐in‐time weak solutions with finite energy to an initial boundary‐value problem and establish the long‐time behaviour of such solutions. The main difficulty is due to the singularity of the gradient magnetic force and the torque. Copyright © 2007 John Wiley & Sons, Ltd.     

The global attractor of g-Navier–Stokes equations with linear dampness on

In this paper, the long time behaviors of g-Navier–Stokes equations with linear dampness on R² were investigated. By using the energy equation method, the existence of the global attractor for the equations was proved without the restriction of the forcing term belonging to some weighted Sobolev space. Moreover, the estimation of the Hausdorff and Fractal dimensions of such attractors were also obtained.     

Global solvability for a large flux of a three‐dimensional time‐dependent Navier–Stokes problem in a straight cylinder

The unique global existence of a solution to nonstationary Navier–Stokes system with prescribed nonzero flux F(t) in an infinite three‐dimensional pipe is proved. The obtained solution remains close to the corresponding nonstationary Poiseuille flow. Moreover, it converges to the Poiseuille flow as |x₃|→∞. Copyright © 2008 John Wiley & Sons, Ltd.     

Trajectory attractor and global attractor for a two-dimensional incompressible non-Newtonian fluid

The authors construct the trajectory attractor and global attractor for an autonomous two-dimensional non-Newtonian fluid.     

Global existence of weak solution to Navier–Stokes equations with large external force and general pressure

We study the 3‐D compressible Navier–Stokes equations with an external potential force and a general pressure. We prove the global‐in‐time existence of weak solutions with small‐energy initial data and with densities being positive and essentially bounded. No smallness assumption is made on the external force. Copyright © 2013 John Wiley & Sons, Ltd.     

Global attractor for a density-dependent sensitivity chemotaxis model

In this paper, a chemotaxi model with reproduction term in a bounded domain Ω ⊂ Rⁿ is discussed. The existence of a global-in-time solution and a global attractor for this model are obtained.