MICROPOLAR FLUID FLOWS WITH DELAY ON 2D UNBOUNDED DOMAINS∗

In this paper, we investigate the incompressible micropolar fluid flows on 2D unbounded domains with external force containing some hereditary characteristics. Since Sobolev embeddings are not compact on unbounded domains, first, we investigate the existence and uniqueness of the stationary solution, and further verify its exponential stability under appropriate conditions – essentially the viscosity δ1 := min{ν, ca + cd} is asked to be large enough. Then, we establish the global well-posedness of the weak solutions via the Galerkin method combined with the technique of truncation functions and decomposition of spatial domain.


Introduction
The micropolar fluid model is firstly derived by Eringen [12] in 1966, which is used to describe the fluids consisting of randomly oriented particles suspended in a viscous medium. The model can be described by the following equations: − (ν + ν r )∆u − 2ν r rotω + (u · ∇)u + ∇p = f, ∇ · u = 0, ∂ω ∂t − (c a + c d )∆ω + 4ν r ω + (u · ∇)ω − (c 0 + c d − c a )∇divω − 2ν r rotu =f , where u = (u 1 , u 2 , u 3 ) is the velocity, ω = (ω 1 , ω 2 , ω 3 ) is the angular velocity field of rotation of particles, p represents the pressure, f = (f 1 , f 2 , f 3 ) andf = (f 1 ,f 2 ,f 3 ) stand for the external force and moments, respectively. The positive parameters ν, ν r , c 0 , c a and c d are the viscosity coefficients. Actually, ν represents the usual Newtonian viscosity and ν r is called the microrotation viscosity. Note that when the gyration is neglected, the micropolar fluid equations are reduced to the classical Navier-Stokes equations. Micropolar fluid model takes an important role in the fields of applied and computational mathematics, there is a wide literature on the mathematical theory of micropolar fluid model (1.1). Here we only illustrate some known results. First, we must mention that Lukaszewicz [15] obtained fruitful results, including the existence and uniqueness of solutions for the stationary problems and the existence of weak and strong solutions for the evolutionary problems, as well as the global existence of solution for the heat-conducting flows and the applications of the micropolar fluids in lubrication theory and in porous media, etc. Also, there are some other papers concentrating on the existence and uniqueness of solutions for the micropolar fluid flows. Galdi and Rionero [13] showed the existence and uniqueness theorems, known in the theorem of the Navier-Stokes equations, are valid for the incompressible micropolar equations too. Yamaguchi [25] established the existence of global strong solution in 3D bounded domain. Boldrini, Durán and Rojas-Medar [2] proved the existence and uniqueness of strong solution in a bounded or unbounded domain Ω ⊆ R 3 having a compact C 2 -boundary. Zhang [28] investigated the global existence and uniqueness of classical solutions to the 2D micropolar fluid flows with fix partial dissipation and angular viscosity. Dong and Zhang [11] proved the global existence and uniqueness of smooth solutions to the 2D micropolar fluid flows with zero angular viscosity on unbounded domains. At the same time, the long time behavior of solutions for the micropolar fluid model has been investigated from various aspects, see, e.g. [7-10, 16, 17, 21, 26, 29]. However, to our knowledge, there are very few articles about the micropolar fluid model with time delay. To date, we have not found in the literature any work that considers the combination of delay terms and unbounded domains.
In the real world, delay terms appear naturally, for instance as effects in wind tunnel experiments (see [18]), in the equations describing the motions of the fluids. The delay situations may also occur, for example, when we want to control the system via applying a force which considers not only the present state but also the history state of the system. There are some articles concerning the pullback asymptotic behavior of solutions to the nonlinear evolution equations with delays on bounded or unbounded domains, see, e.g. [3-6, 14, 20, 23, 24].
In this paper, we consider the situation that the velocity component in the x 3direction is zero and the axes of rotation of particles are parallel to the x 3 axis. That is, u = (u 1 , u 2 , 0), ω = (0, 0, ω 3 ), f = (f 1 , f 2 , 0),f = (0, 0,f 3 ), g = (g 1 , g 2 , 0) and g = (0, 0,g 3 ). Let Ω ⊆ R 2 be an open set with boundary Γ that is not necessarily bounded but satisfies the following Poincaré inequality: There exists λ 1 > 0 such that λ 1 ϕ 2 Then, we discuss the following equations of 2D non-autonomous incompressible micropolar fluid flows: , ω(0, ·)) = (u 0 (·), ω 0 (·)), (u(t, ·), ω(t, ·)) = (φ 1 (t, ·), φ 2 (t, ·)), t ∈ (−h, 0), x ∈ Ω, where T > 0 is given,ᾱ := c a + c d , x := (x 1 , x 2 ) ∈ Ω, (u 0 , ω 0 ) is the initial velocity filed, g andg stand for the external force containing some hereditary characteristics u t and ω t , which are defined on [−h, 0] as follows: In addition, (φ 1 , φ 2 ) represents the initial datum in the interval of time (−h, 0), where h is a positive fixed number, and For the sake of convenience, we introduce the following useful operators: (1.5) whereν = (ν +ν r ) and the notation V will be defined later. Then, we can formulate the weak version of equations (1.3) as follows: where w := (u, ω), F (t) = F (t, x) := (f,f ) and G(t, w t ) := (g(t, u t ),g(t, ω t )). There are two goals in writing this thesis. The first goal is to prove the existence and uniqueness of the stationary solution and to verify its exponential stability, exactly, we reveal that when the viscosity δ 1 := min{ν, c a + c d } is large enough, the weak solution of the evolutionary system (1.6) exponentially approaches the stationary solution as time increasing infinity. In this part, we need pay enough attention and give careful analysis for each term. In addition, the delay term will also increase the difficulty of the estimates.
The second goal is to establish the global well-posedness of the weak solution of system (1.6). Due to the lack of compact embedding in an unbounded domain, which will result in some obstacles in the process of using the classical Galerkin method to prove the existence of solutions. To overcome this difficulty, we utilize the Galerkin method combined with the technique of truncation function and the decomposition of spatial domain, and classical method to complete our purpose.
It is worth to mentioning that the existence and uniqueness of the weak solutions for the Navier-Stokes with delay on smooth bounded domains has been established by Caraballo and Real in [3]. Later, they investigated the asymptotic behaviour of the weak solutions in [4]. Afterwards, Garrido-Atienza and Marín-Rubio in [14] extended the results of [3] to unbounded domains. Moreover, they studied the existence and uniqueness of the stationary solution and its stability. We want to point out that the main idea of this paper originates from paper [14,20,27].
Compared with the Navier-Stokes equations studied in [14], the angular velocity field ω of the micropolar particles in the micropolar fluid flows leads to a different nonlinear term B(u, w) and an additional term N (w) in the abstract equation (see (1.6)). For that reason, more delicate estimates and analysis are required in our studies.
Throughout this paper, we denote the usual Lebesgue space and Sobolev space (see [1]) by L p (Ω) and W m,p (Ω) endowed with norms · p and · m,p , respectively.  where · H , · V , · H and · V are defined by (·, ·)− the inner product in L 2 (Ω), H or H, ·, · − the dual pairing between V and V * or between V and V * . Throughout this article, we simplify the notations · 2 , · H and · H by the same notation · if there is no confusion. Furthermore, we denote The rest of this paper is organized as follows. In section 2, we first make some preliminaries. Then, we concentrate on establishing the existence and uniqueness of the stationary, and further verifying its exponential stability, that is, under suitable conditions, the weak solution, the existence of which could be ensured by section 3, exponentially approaches the stationary solutions as time goes to +∞. In section 3, we show the global well-posedness of the weak solutions.

Stability of stationary solutions
We divide this section into two subsections. In the first subsection, we make some necessary preliminaries. In the other subsection, we prove the stability of the stationary solutions.

Preliminaries
To begin with, let us give some useful properties and estimates about the operators defined in (1.5). That is, (1) The operator A is linear continuous both from V to V * and from D(A) to H, and so is for the operator N (·) from V to H, where D(A) := V ∩ (H 2 (Ω)) 3 .
(2) The operator B(·, ·) is continuous from V × V to V * . Moreover, for any u ∈ V and w ∈ V , there holds Proof.
(1) The continuity of the operators A and N (·) can be deduced directly from their definition. The linearity of the operator A is evident. So we only need check the linearity of the operator N (·). Indeed, for any (2) The continuity of the operator B(·, ·) can be also obtained from its definition. Next, we verify (2.1). In fact, for any u ∈ V, w ∈ V , we have Hence, (2.1) is valid as a consequence of (2.2). The proof is complete. We further have Lemma 2.2 (see [16,19,26]).
(1) There are two positive constants c 1 and c 2 such that (2) There exists some positive constant λ 0 which depends only on Ω, such that for In addition, where δ 1 := min{ν,ᾱ}.
Next, we recall a key lemma from [14] as follows.
where Π h f is the translation function: Then F is precompact in L r (I; E).
Finally, we end this subsection with the definition of weak solution of (1.6).
holds in the distribution sense of D (0, T ).

Stability of stationary solutions
In this subsection, we prove the existence and uniqueness of stationary solutions to the micropolar fluid flows provided the viscosity is large enough, when the delay term has a special form. Furthermore, in a little stronger conditions, we verify its exponential stability. From now on, we suppose that ρ(·) (2.7) In the following, we concentrate on establishing the existence and uniqueness of the stationary of (1.6). That is, to find a function w * = (u * , ω * ) ∈ V such that (1) there exists at least one solution to (2.8), (2) under the extra condition: λ Where the constant λ 0 comes from (2.4).
Proof. (1) Existence of stationary solutions. Firstly, we take an orthonormal , v m } and consider the following problem: , v m . On one hand, it is not difficult to check that, for a fixed q m = (p m , q m (3) ) ∈ V m , the function σ(·, ·) is bilinear, continuous and coercive in V m × V m . On the other hand, the function v m → F, v m + (G(q m ), v m ) is obviously linear and continuous. Therefore, by Lax-Milgram theorem, for each fixed q m = (p m , q m (3) ) ∈ V m , there exists a unique solution to problem (2.9), which we denote w m . Then, consider the map E m (·) : V m → V m , which is defined by Next, we are devoted to proving that, for each m, there exists at least one fixed point of the mapping E m (·). This implies that there exists a w m ∈ V m satisfying In order to proceed, taking v m = w m in (2.9), and using (1.2), (2.2), (2.6) and (2.7), we deduce which is a convex compact set of V . Then, it follows from (2.12) that E m (·) maps K m to K m . In the following, we are going to apply the Brouwer fixed point theorem to E m (·) Km . For this end, it only remains to show that E m is continuous. Indeed, take q m i ∈ K m , i = 1, 2, and denote w m )) the respective solutions of (2.9). Then, it holds that with the aid of (1.2), (2.2), (2.4), (2.6) and (2.7), the above inequality gives where we also used Hölder inequality and the facts q m The inequality (2.13) implies E m is continuous. At this stage, we can conclude that, for each m ∈ N, there exists a w m = (u m , ω m ) ∈ V m satisfying (2.10).
Finally, we will pass to the limit in (2.10) to obtain the existence of solutions of (2.8). Similar to (2.11), taking v m = w m in (2.10), we obtain (2.14) So, we may extract a subsequence (denoting by the same symbol) {w m } such that Moreover, for any regular bounded set Q ⊂ Ω, we have the same uniform bounds of w m Q , which means, using the compact injection, that w m Q → w Q strongly in (L 2 (Q)) 3 .
(2.16) Based on the above argument, for a fixed v j ∈ {v j } ∞ j=1 , denote by Q j the support of v j (which is compact) and take Q ⊂ Ω a bounded open set with smooth boundary containing Q j , then we not only have the weak convergence w m w in V (Q j ) but the strong convergence w m → w in (L 2 (Q j )) 3 . Furthermore, it holds that At last, combining with (2.15), we may pass to the limit with respect to m for every term in (2.10) to obtain Since span{v 1 , v 2 , · · · , v n , · · · } is dense in V , we conclude that there exists at least one function w * := w satisfies (2.8).
(2) Uniqueness of the stationary solution. Now, we prove uniqueness of solution to (2.8) under the extra condition λ 1 2 Suppose there are two solutions w 1 , w 2 to (2.8). Taking the difference, it holds that In particular, taking v = w 1 − w 2 , similar to (2.13), we have where, in the second inequality, we have used (2.14) and the fact u 1 − u 2 V w 1 − w 2 V . It follows from the above inequality that Hence, the uniqueness follows as long as λ This completes the proof.
Next, under a little stronger condition than that in Theorem 2.1, which ensures the existence and uniqueness of the stationary solution w * of (2.8), we prove that the weak solution of the evolutionary problem (1.6) exponentially approaches w * as time increases to infinity. That is, the following theorem.

Global well-posedness of the weak solutions
In this section, we concentrate on proving the global existence, uniqueness and stability of the weak solution to system (1.6).
In order to establish the global well-posedness of the weak solutions, the following assumption is required. t → G(t, ξ) ∈ (L 2 (Ω)) 3 is measurable.
Now, we show the existence of the weak solutions in the following theorem.
Proof. We will divide the proof into three steps.
Step One: Local existence and uniqueness of the Galerkin approximate solutions.
Denote V m :=span{v 1 , v 2 , · · · , v m } and consider the projector where the coefficients β m,j (t) are desired to satisfy the following Cauchy problem of ordinary differential equations: Based on Theorem 2.1 in [14], the existence and uniqueness of the Galerkin approximate solution follows.
Step Two: A priori estimates of the Galerkin approximate solutions.
We now deduce a priori estimates to obtain the global existence of the Galerkin approximate solutions. Multiplying (3.1) 1 by β m,j (t), summing up for j from 1 to m and using (2.2) and (2.6), we have Multiplying (3.2) by e γt , we obtain Replacing the time variable t with θ in the above inequality, then integrating it for θ over [0, t] gives By Young's inequality and assumption (A), we see that

.4) and (3.5) into (3.3), we have
It is obtained easily from the above inequality that there exist two constants k 1 and k 2 (depending on w 0 , φ, δ 1 , F, G, h, T , but not on m nor t * T ) such that Moreover, observe that w m = P m φ in (−h, 0) converges to φ in L 2 (−h, 0; V ). Thus, we can take t * = T and obtain that which together with the local existence obtained in step one gives to the global existence of the Galerkin approximate solution for all time t ∈ [0, T ].
Step Three: Existence of the global weak solutions. We will prove that the limit function of the Galerkin approximate solutions is a weak solution of (1.6). Using the diagonal procedure, we deduce from (3.7) that there exists a subsequence (which is still denoted by) {w m }, an element w ∈ L ∞ (0, T ; H) ∩ L 2 (0, T ; V ) such that w m * w weakly star in L ∞ (0, T ; H) as m → ∞, w m w weakly in L 2 (0, T ; V ) as m → ∞. For the sake of clarity, we give the proof of (3.9) in the back of the present theorem. Now, fix an element v j and let ϕ ∈ C 1 ([0, T ]) with ϕ(T ) = 0. Then, it follows from (3.1) that In the following, we are committed to passing to the limit in (3.10) to obtain a weak solution. Choose a subsequence (denoted again by w m ) by using diagonal procedure that satisfies (3.9) for a sequence of regular bounded open sets Q j ⊂ Ω containing all supports of functions v j of the basis. Observe that for every v ∈ V , by the density, there exists a sequence {v j } ⊂ V such that v j → v in V as j → ∞. Namely, for any > 0, there exists a n > 0 such that Thus, from (3.8), (3.9) and assumption (A), it is not difficult to see that for the above , there exists a m n such that, for all m m , Similarly, it holds that In addition, by (1.5), (2.4), (2.5), (3.7)-(3.9), (3.11) and Lemma 2.1, we have where C i , i = 1, 2, 3, 4, 5, 6, 7 are positive constants. According to the above seven inequalities and the arbitrariness of , we can pass to the limit in (3.10) and obtain Since (w(t), v j )ϕ(t) ∈ H 1 (0, T ), we also can obtain an analogous expression to (3.12) with (w(0), v) instead of (w 0 , v). This implies (w(0)−w 0 , v) = 0 for all v ∈ V , hence w(0) = w 0 . It makes sense at time t = 0. Writing (3.12) for ϕ ∈ D(0, T ), w satisfies (1.6) in the distribution sense. This completes the proof.
(2) For a general Q ⊂ Ω the above comment may be not true since Q and Ω can share part of their boundaries. The compact injection from H 1 may not hold for lack of regularity on the boundary Γ, but it does in H 1 0 . Therefore, we consider a truncation argument.
Obviously, the condition (ii) in Lemma 2.3 follows from (3.7). In the following, we concentrate on verifying the condition (i). In fact, we will prove that, for the whole domain Ω, Consider h > 0 arbitrarily small. From (3.1), we see that, for (t, t + h) ⊂ (0, T ), Multiplying the above inequality by β m,j (t + h) − β m,j (t) and summing in j, we obtain (3.14) In the following, we estimate the terms on the right-half side of (3.14) one by one. First, according to the definitions of the operators in (1.5), Lemma 2.1, Lemma 2.2, and using Hölder inequality, we have (3.16) and Next, it is easy to know from Schwartz's inequality that (3.18) Finally, using (1.2) and Schwartz's inequality, we obtain Now, substituting (3.15)-(3.19) into (3.14) and integrating the resultant inequality from 0 to T − h, we have Then, since 0 θ − θ − h h, with the aid of (3.6) and the Fubini theorem, we obtain From (3.7) and the assumption (A), it is not difficult to conclude that T 0 H m (θ)dθ is bounded. Therefore, (3.13) is valid, that is, the condition (i) in Lemma 2.3 holds. Consequently, with the help of Lemma 2.3, (3.9) follows.
Remark 3.2. The existence and uniqueness of the stationary solution of (2.8) and the existence of the weak solutions to system (1.6) can be obtained in the same way in 3D unbounded domains. The key is the nonlinear term B(u, w), the estimates (which can be deduced by Hölder and Gagliardo-Nirenberg inequality) for this term is different in the cases fo 2D and 3D domains. For more detail, one can refer to [22, §2, §3 and §10].
Remark 3.3. Existence, uniqueness and stability of the solution have been established under different conditions -essentially the viscosity δ 1 := min{ν,ᾱ} is asked to be large enough. We also want to point out that there still much work to be done concerning the micropolar fluid flows with delay on unbounded domains. For example, we could study the attractors, further, investigate the regularity, boundedness and tempered behavior of the pullback attractors. These issues will be the topics of some other papers.