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.     

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).     

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 heat convection problem in a micropolar fluid

The article is devoted to describe asymptotics in the heat convection problem for a micropolar fluid in two dimensions. We show the existence and the uniqueness of global in time solutions and then prove the existence of a global attractor for considered model. Next, the Hausdorff dimension of the global attractor is estimated. Copyright © 2006 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.     

A regularity criterion for 3D micropolar fluid flows

In this work, we prove a regularity criterion for micropolar fluid flows in terms of the pressure in Besov space.     

Uniform attractor for 2D magneto-micropolar fluid flow in some unbounded domains

We investigate the flow of a magneto-micropolar fluid in an arbitrary unbounded domain on which the Poincaré inequality holds. Assuming homogeneous boundary conditions and the external fields to be almost periodic in time we prove the existence of the uniform attractor by using the energy method [10] which we generalize to nonautonomous systems. We consider the problem in an abstract setting that allows to include also other hydrodynamical models. In particular, we extend the result of R. Rosa [12] from autonomous to nonautonomous Navier-Stokes equations in unbounded domains.     

Uniform attractor for 2D magneto-micropolar fluid flow in some unbounded domains

We investigate the flow of a magneto-micropolar fluid in an arbitrary unbounded domain on which the Poincaré inequality holds. Assuming homogeneous boundary conditions and the external fields to be almost periodic in time we prove the existence of the uniform attractor by using the energy method [10] which we generalize to nonautonomous systems. We consider the problem in an abstract setting that allows to include also other hydrodynamical models. In particular, we extend the result of R. Rosa [12] from autonomous to nonautonomous Navier-Stokes equations in unbounded domains.     

On the regularity criterion for three-dimensional micropolar fluid flows in Besov spaces

This paper studies the regularity criterion of weak solutions for three-dimensional (3D) micropolar fluid flows. When the velocity field satisfies for −1<r<1, then the weak solution (u,w) is regular on (0,T]. The methods are mainly based on the Fourier localization technique and Bony’s para-product decomposition.     

On exact solutions for thin film flows of a micropolar fluid

The present paper contains the exact solutions for thin film flows of a micropolar fluid. Thin films are considered for three different geometries namely: (i) flow down an inclined plane, (ii) flow on a moving belt and (iii) flow down a vertical cylinder. The exact solution is obtained for the velocity and micro-rotation in first two cases for both weak and strong concentrations of the microelements. However, the exact expression for the micro-rotation is obtained for the third case and a numerical solution is presented for the velocity. The micro-rotation and velocity fields are sketched and discussed for different values of the material parameters. It is observed that the rotation of the microelements at the boundary increases the velocity when compared with the case when there is no rotation at the boundary.     

Vortex generated fluid flows in multiply connected domains

A fluid flow in a multiply connected domain generated by an arbitrary number of point vortices is considered. A stream function for this flow is constructed as a limit of a certain functional sequence using the method of images. The convergence of this sequence is discussed, and the speed of convergence is determined explicitly. The presented formulas allow for an easy computation of the values of the stream function with arbitrary precision in the case of well-separated cylinders. The considered problem is important for applications such as eddy flows in oceans. Moreover, since finding the stream function of the flow is essentially identical to finding the modified Green’s function for Laplace’s equation, the presented method can be applied to a more general class of applied problems which involve solving the Dirichlet problem for Laplace’s equation.     

Pullback attractors for nonautonomous reaction-diffusion equations in unbounded domains

In this paper, the existence of (L²(Rn),L²(Rn))-pullback attractors and (L²(Rn),H¹(Rn))-pullback attractors are proved for reaction-diffusion equation in unbounded domains.     

Gradient estimates for elliptic systems in non-smooth domains

We obtain the global W ^1,p , 1 < p < ∞, estimate for the weak solution of an elliptic system with discontinuous coefficients in non-smooth domains without using maximal function approach. It is assumed that the boundary of a bounded domain is well approximated by hyperplanes at every point and at every scale, and that the tensor coefficients belong to BMO space with their BMO semi-norms sufficiently small. S.-S. Byun was supported in part by KRF-2006-C00034 and L. Wang was supported in part by NSF Grant 0701392.     

Vector potentials in three-dimensional non-smooth domains

This paper presents several results concerning the vector potential which can be associated with a divergence-free function in a bounded three-dimensional domain. Different types of boundary conditions are given, for which the existence, uniqueness and regularity of the potential are studied. This is applied firstly to the finite element discretization of these potentials and secondly to a new formulation of incompressible viscous flow problems.     

Vector-valued Hardy spaces in non-smooth domains

We characterize the Radon-Nikodým property of a Banach space X in terms of the existence of non-tangential limits of X-valued harmonic functions u defined in a domain D⊂Rn, n>2, with Lipschitz boundary and belonging to maximal Hardy spaces. This extends the same result previously known for the unit disk of C. We also prove an atomic decomposition of the Borel X-valued measures in ∂D that arise as boundary limits of X-valued harmonic functions whose non-tangential maximal function is integrable with respect to harmonic measure of ∂D.     

Pullback attractors for the non-autonomous Benjamin-Bona-Mahony equation in unbounded domains

The existence of a pullback attractor is proven for the non-autonomous Benjamin-Bona-Mahony equation in unbounded domains.The asymptotic compactness of the solution operator is obtained by the uniform estimates on the tails of solutions.     

1-D compressible viscous micropolar fluid model with non-homogeneous boundary conditions for temperature: A local existence theorem

We consider non-stationary 1-D flow of a compressible viscous and heat-conducting micropolar fluid, assuming that it is in thermodynamical sense perfect and polytropic. The homogeneous boundary conditions for velocity and microrotation, as well as non-homogeneous boundary conditions for temperature are assumed. Using the Faedo-Galerkin method we prove a local-in-time existence of a generalized solution.