Long-time behavior of micropolar fluid equations in cylindrical domains
Introduction
The study of attractors is an important part of examining dynamical systems. It was thoroughly investigated in many works (see e.g. [1], [2], [3], [4], [5], [6]). These classical results cover many both autonomous and non-autonomous equations of mathematical physics. Roughly speaking, the general abstract theory justifying attractors’ existence may be applied when the equations describing autonomous systems possess unique, strong and global solutions. For non-autonomous systems some additional uniformity for external data is required. Further obstacles arise when unbounded domains are examined (see e.g. [7], [8] and references therein).
In three dimensions the situation for nonlinear evolution problems is even more complex. In the case of micropolar equations we lack information about the regularity of weak solutions at large. Thus, results concerning the existence of uniform attractors are merely conditional (see e.g. [5]). In this article we present some remedy which is based on [9]. It takes into account that global and strong solutions exist if some smallness of -norms on the rate of change of the external and the initial data is assumed (see [10]). This leads to a restriction of the uniform attractor to a proper phase space.
Introduced by A. Eringen in [11], the micropolar fluid equations form a useful generalization of the classical Navier–Stokes model in the sense that they take into account the structure of the media they describe. With many potential applications (see e.g. [12], [13], [14], [15]) they became an interesting and demanding area of interest for mathematicians and engineers (see e.g.[11], [16], [17], [18], [15], [19] et al.).
In this article we study the following initial–boundary value problem where . By we mean a cylindrical type domain, i.e. where and are real constants, is a closed curve of class . The functions and denote the velocity field, the micropolar field and the pressure, respectively. The external data (forces and momenta) are represented by and . The viscosity coefficients and are real and positive.
The choice of the boundary conditions for and was thoroughly described in [10]. Note that usually the zero Dirichlet condition is assumed for both the velocity and the micropolar field. From a physical point of view this is not suitable for some classes of fluids (see e.g. [20]). In fact, the explanation of particle behavior at the boundary is far more complex and was extensively discussed in [21], [22], [23], [24].
The article is organized as follows: in the next section we present an overview of the current state of the art. Subsequently we introduce notation and give some preliminary results. In Section 4 the main results are formulated. Section 5 is entirely devoted to the basics of semi-processes. In Sections 6 Existence of the uniform attractor, 7 Convergence to stationary solutions for large, 8 Continuous dependence on modeling we prove the existence of the restricted uniform attractor and analyze the behavior of the micropolar fluid flow for the large viscosity . We demonstrate that for time independent data the global and unique solution converges to the solutions of the stationary problem and if then the trajectories of micropolar flow and of the Navier–Stokes flow differ indistinctively.
Section snippets
State-of-the-art
The study of the uniform attractors has mainly been focused on the case of unbounded domains (see [25], [26] and the references therein) which is barely covered by general abstract theory introduced in [5]. Note that both these studies cover only the two dimensional case. In three dimensions no results have been obtained so far due to lack of regularity of weak solutions. Another obstacle arises from relaxing the assumption of certain uniformity of the external data. To avoid these
Notation and preliminary results
In this paper we use the following function spaces:
- •
, where , is the closure of in the norm
- •
, where , is simply ,
- •
, where , is the closure of in the norm
- •
is the set of all smooth solenoidal functions whose normal component vanishes on the boundary
- •
is the closure of in ,
- •
,
- •
is the
Main results
The main result of this paper reads as follows. Theorem 2 Let the assumption from Theorem 1 be satisfied. Suppose thatfor anyand, whereis any element of the set. Then, the family of semi-processescorresponding to problem (1.1) has a uniform attractorwhich coincides with the uniform attractorof the family of semiprocesses, i.e.
Basics of semiprocesses
We begin with recalling a few facts from [4, Ch. 1] and [5, Ch. 2].
Let be a semigroup acting on a complete metric or Banach space . Denote by the set of all bounded sets in with respect to metric in . We say that is an attracting set for if for any
Now we may define an attractor as follows. Definition 5.1 A set is called a global attractor for the semigroup , if it satisfies: is compact in , is an attracting set for , is strictly
Existence of the uniform attractor
In this section we prove the existence of the uniform attractor to problem (1.1) restricted to . By we denote the family of all bounded sets of . We begin by introducing some fundamental definitions. Definition 6.1 A family of processes is said to be uniformly bounded if for any the set
Definition 6.2 A set is said to be uniformly absorbing for the family of processes , if for any and for every there exists
Convergence to stationary solutions for large
In this section we prove Theorem 3.
Proof The existence of solutions for large and their estimate follows, after slight modifications, from [34] and [15, Ch. 2, Section 1, Theorems 1.1.1–1.1.3]. Let and . Then satisfies Multiplying the first and the third equation by
Continuous dependence on modeling
In this section we examine the difference between and the solution to problem (4.2).
Observe, that (4.2) is the same as (1.1) with . Therefore, for close to zero we may measure in some sense the deviation of the flows of micropolar fluids from that of modeled by the Navier–Stokes equations. This problem was considered locally in [35] and globally in [19, Section 5].
Proof of Theorem 4 Let us denote . Then the pair is a solution to the problem
Acknowledgments
The author was financially supported by the Ministry of Science and Higher Education, under project number N N201 393137.
The author wishes to thank the referee for her/his help in improving the paper.
References (37)
- et al.
-uniform attractor and asymptotic smoothing effect of solutions for a nonautonomous micropolar fluid flow in 2d unbounded domains
Nonlinear Anal. RWA
(2008) Large time existence of strong solutions to micropolar equations in cylindrical domains
Nonlinear Anal. RWA
(2013)Micropolar Fluids. Theory and Applications
(1999)- et al.
Magneto-micropolar fluid motion: global existence of strong solutions
Abstr. Appl. Anal.
(1997) - et al.
Velocity distribution and other characteristics of steady and pulsatille blood in fine glas tubes
Biorheology
(1970) - et al.
Pullback attractors of non-autonomous micropolar fluid flows
J. Math. Anal. Appl.
(2007) Pullback attractor for non-homogeneous micropolar fluid flows in non-smooth domains
Nonlinear Anal. RWA
(2009)- et al.
On -pullback attractors for nonautonomous micropolar fluid equations in a bounded domain
Nonlinear Anal. TMA
(2009) - et al.
Comparison between trajectory and global attractors for evolution systems without uniqueness of solutions
Internat. J. Bifur. Chaos Appl. Sci. Engrg.
(2010) - B. Nowakowski, Global existence of strong solutions to micropolar equations in cylindrical domains, ArXiv e-prints,...
On global regular solutions to the Navier–Stokes equations in cylindrical domains
Topol. Methods Nonlinear Anal.
Global Attractors in Abstract Parabolic Problems
Attractors for Equations of Mathematical Physics
Uniform attractor for 2d magneto-micropolar fluid flow in some unbounded domains
Z. Angew. Math. Phys.
Cited by (8)
Pullback exponential attractors for the non-autonomous micropolar fluid flows
2018, Acta Mathematica ScientiaMicropolar fluid flows with delay on 2D unbounded domains
2018, Journal of Applied Analysis and ComputationAsymptotic behavior of pullback attractors for non-autonomous micropolar fluid flows in 2D unbounded domains
2018, Electronic Journal of Differential EquationsPullback dynamical behaviors of the non-autonomous micropolar fluid flows with minimally regular force and moment
2018, Communications in Mathematical SciencesH<sup>2</sup> -boundedness of the pullback attractor of the micropolar fluid flows with infinite delays
2017, Boundary Value ProblemsGlobal well-posedness and pullback attractors for a two-dimensional non-autonomous micropolar fluid flows with infinite delays
2017, Communications in Mathematical Sciences