On upper and lower bounds of higher order derivatives for solutions to the 2D micropolar fluid equations

J. Math. Anal. Appl. 334 (2007) 1386–1399

www.elsevier.com/locate/jmaa

On upper and lower bounds of higher order derivativesfor solutions to the 2D micropolar fluid equations

Bo-Qing Dong a,∗, Zhi-Min Chen b

a School of Mathematical Sciences, Nankai University, Tianjin 300071, PR Chinab School of Engineering Sciences, Ship Science, University of Southampton, Southampton SO17 1BJ, United Kingdom

Received 20 May 2006

Available online 25 January 2007

Submitted by M.C. Nucci

Abstract

The present paper is concerned with asymptotic behaviours of the solutions to the micropolar fluid motionequations in R2. Upper and lower bounds are derived for the L2 decay rates of higher order derivatives ofsolutions to the micropolar fluid flows. The findings are mainly based on the basic estimates of the linearizedmicropolar fluid motion equations and generalized Gronwall type argument.© 2007 Elsevier Inc. All rights reserved.

Keywords: Micropolar fluid motion equations; Upper and lower bounds; L2 decay

1. Introduction

Consider the two-dimensional mathematical model of the incompressible viscous flows gov-erned by the micropolar fluid equations

∇ · v = 0,

∂

∂tv − (ν + κ)�v − 2κ∇ × w + ∇π + v · ∇v = 0,

∂

∂tw − γ�w + 4κw − 2κ∇ × v + v · ∇w = 0

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

(1.1)

* Corresponding author.E-mail address: [email protected] (B.-Q. Dong).

0022-247X/$ – see front matter © 2007 Elsevier Inc. All rights reserved.doi:10.1016/j.jmaa.2007.01.047

B.-Q. Dong, Z.-M. Chen / J. Math. Anal. Appl. 334 (2007) 1386–1399 1387

associated with the initial conditions

v|t=0 = v0, w|t=0 = w0. (1.2)

Here v = (v1, v2), π and w denote the unknown velocity vector field, the scalar pressure field andthe scalar gyration field over the fluid-time domain R2 × (0,∞). ν > 0 is a kinematic viscositycoefficient, κ � 0 and γ > 0 are gyration viscosity coefficients, and

∇ × v = ∂v2

∂x1− ∂v1

∂x2, ∇ × w =

(∂w

∂x2,− ∂w

∂x1

).

This micropolar fluid motion model, a special model of microfluid motions, was introduced byEringen [10] and exhibits micro-rotational effects and micro-rotational inertia. Physically it mayrepresent adequately the fluids consisting of bar-like elements. Certain anisotropic fluids, e.g. liq-uid crystals made up of dumbbell molecules, are of the type. When the scalar gyration vector fieldis neglected, the micropolar fluid motion model reduces to the incompressible Navier–Stokesequations (refer to [16,30]).

There is a large literature on the mathematical theory of micropolar fluid equations. The exis-tence and uniqueness problems were extensively studied by Lange [17], Galdi and Rionero [11],Sava [25], and recently, by Łukaszewicz [19], Resndiz and Rojas-Medar [24] and referencestherein. As for the dynamical behavior of (1.1)–(1.2), one may also refer to the study by Chenand Dong [5,6], Łukaszewicz [20,21].

The time decay rate estimates of the Navier–Stokes equations were originally from the cele-brated work of Leray [18]. Algebraic time decay properties of weak solutions were successfullyderived by Schonbek [26–28], Kajikiya and Miyakawa [14] and Wiegner [31]. Especially, byapplying a Fourier splitting method, Schonbek [26] obtained the following upper–lower boundresult

c(1 + t)−n4 �

∥∥u(t)∥∥ � c1(1 + t)−

n4 if

∫u0 dx �= 0.

By assuming some other nonzero condition on the initial velocity field u0, she [27] improved

the decay rates described as (1 + t)− n+24 . Furthermore, by a combination of the Fourier splitting

method and Lq decay estimates (see Kato [15]), Schonbek [28] also established the decay esti-mates of higher order derivatives of solutions to the two-dimensional Navier–Stokes equationsexpressed as∥∥∇mu(t)

∥∥ � c(1 + t)−m+1

2 ,

where ∇m represents any differential operator in the form (∂x1)α1(∂x2)

α2 for α1 +α2 = m. More-over, based on so-called Gevrey estimates, Oliver and Titi [23] studied the upper and lowerbounds of higher order derivatives of Navier–Stokes equations in Rn with the aid of the L2 de-cay rate estimates

c5(1 + t)−m+γ0

2 �∥∥∇mu(t)

∥∥ � c6(1 + t)−m+γ0

2 ,

where γ0 is the decay rate of the linear heat equation solution.The interested readers may also refer to [1–4,12,13,22] for other interesting time decay prob-

lems of Navier–Stokes equations and [7–9] for time decay problems of non-Newtonian flows.However, not much work on the time decay problem of the two-dimensional micropolar fluid

motion equations is published. Moreover, compared with the Navier–Stokes equations, an extra

1388 B.-Q. Dong, Z.-M. Chen / J. Math. Anal. Appl. 334 (2007) 1386–1399

difficulty arises in the examination of the micropolar fluid motion model due to the occurrenceof the gyration vector field.

The objective of this paper is to show the upper–lower bound estimates for the L2 decayrates of higher order derivatives of solutions to the micropolar fluid motion equations. To deducethe estimates, we examine the decay estimates of derivatives for the solutions to the linearizedmicropolar fluid motion equations and then extend the estimates on linearized equations to thenonlinear equations by using a generalized Gronwall type argument.

To simplify the form of (1.1), we denote

u =⎛⎝v1

v2

w

⎞⎠ , u0 =

(v0

w0

)=

⎛⎝v10

v20

w0

⎞⎠ , D =

⎛⎝ ∂1

∂2

0

⎞⎠ ,

and apply formally the operator ∇�−1∇· to the second equation of (1.1) to obtain the expressionof the pressure force

∇π = −∇�−1∇ · (v · ∇v).

We thus introduce a new operator in the form

Pu = u − D�−1D · u,

which is a bounded projection mapping from the Lebesgue space Lr(R2) onto the space

Lrσ

(R2) = {

u ∈ Lr(R2); D · u = 0 in the sense of distributions

}, 1 < r < ∞,

due to Lr -estimate with respect to the Laplace operator �.With the use of these expressions, we formulate (1.1) in the following dynamical system

∂

∂tu + Au + P(v · ∇u) = 0, (1.3)

where A is the self-conjugate and linear differential operator

A =⎛⎝−(ν + κ

2 )� 0 −κ∂2

0 −(ν + κ2 )� κ∂1

κ∂2 −κ∂1 (2κ − γ�)

⎞⎠ .

The linearized micropolar flow is now governed by the system

∂

∂tu + Au = 0, (1.4)

and is expressed in the form as a semigroup e−tAu0 by taking into account the following initialvalue condition:

u(0, x) = u0(x). (1.5)

With the use of the notation described in Section 2, we can now summarize the main result ofthis paper as follows.

Theorem 1.1. Suppose that u(t) is a solution of micropolar fluid motion equations (1.1)–(1.2)with initial vector field u0 ∈ Hm (m � 0) satisfying

ρ(r) ≡2π∫ ∣∣u0(rθ)

∣∣dθ = cr2γ−2 + o(r2γ−2) as r → 0, (1.6)

0


for a constant 1 < γ < 2, then there exist two positive constants c and c1 such that

c(1 + t)−m+γ

2 �∥∥∇mu(t)

∥∥2 � c1(1 + t)−

m+γ2 , for large t. (1.7)

It should be mentioned that the upper and lower bounds of solutions here are optimal in thesense that they coincide with the decay rates of the heat flow e�tu0.

This paper is organized as follows. In Section 2, we investigate the basic estimates for the lin-earized micropolar fluid motion equations. In Section 3, we present the auxiliary decay estimatesof micropolar fluid motion equations. Section 4 is devoted to the proof of upper bound describedTheorem 1.1, whereas Section 5 is contributed to the derivation of the lower bound expressed inTheorem 1.1.

2. Estimates on the linearized equations

We use the following usual notation:

‖ · ‖p = the norm of the Lebesgue space Lp(R2),

‖ · ‖ = the norm ‖ · ‖2,

Hk = the Hilbert space{u ∈ L2(R2); ∥∥∇ku

∥∥ < ∞}.

The Fourier transformation of a function f is denoted by

Fu(ξ) = u(ξ) =∫R2

e−ix·ξ udx, i = √−1,

and c, c1, c2, . . . , are generic positive constants which may only depend on the initial data u0and may vary from line to line.

Letting the matrix

Λ =⎛⎜⎝

(ν + κ2 )|ξ |2 0 −κiξ2

0 (ν + κ2 )|ξ |2 κiξ1

κiξ2 −κiξ1 (2κ + γ |ξ |2)

⎞⎟⎠ ,

and applying the Fourier transform to (1.4)–(1.5), we have

∂

∂tu + Λu = 0, u(0) = u0, (2.1)

or the semigroup

Fe−Atu0 = e−Λt u0.

We now present upper–lower bound estimates with respect to the semigroup.

Lemma 2.1. Let σ1 = 1/2 min{ν, γ }, σ2 = 2(max{ν, γ } + κ/2). Then the estimates

e−σ2|ξ |2t |u0| �∣∣e−Λt u0

∣∣ � e−σ1|ξ |2t |u0|, (2.2)∥∥∇me−Atu0∥∥ � ct−

m2 − 1

2 ‖u0‖1, (2.3)∥∥∇e−Atu0∥∥ � ct−

34 ‖u0‖ 4

3(2.4)

hold true.


Proof. Let us begin with the derivation of the eigenvalues of the matrix Λ or the roots of thefollowing spectral polynomial:

0 =

∣∣∣∣∣∣∣(ν + κ

2 )|ξ |2 − λ 0 −κiξ2

0 (ν + κ2 )|ξ |2 − λ κiξ1

κiξ2 −κiξ1 (2κ + γ |ξ |2) − λ

∣∣∣∣∣∣∣=

((ν + κ

2

)|ξ |2 − λ

)(((ν + κ

2

)|ξ |2 − λ

)(2κ + γ |ξ |2 − λ

) − κ2ξ21

)

− κ2ξ22

((ν + κ

2

)|ξ |2 − λ

)

=((

ν + κ

2

)|ξ |2 − λ

)(((ν + κ

2

)|ξ |2 − λ

)(2κ + γ |ξ |2 − λ

) − κ2|ξ |2)

=((

ν + κ

2

)|ξ |2 − λ

)

×{λ2 −

((2κ + γ |ξ |2) +

(ν + κ

2

)|ξ |2

)λ + 2νκ|ξ |2 + γ

(ν + κ

2

)|ξ |4

}.

We thus have the three different eigenvalues

λ1 =(

ν + κ

2

)|ξ |2,

λ2 = 1

2

((2κ + γ |ξ |2) +

(ν + κ

2

)|ξ |2

)

− 1

2

√((2κ + γ |ξ |2) +

(ν + κ

2

)|ξ |2

)2

− 4

(2νκ|ξ |2 + γ

(ν + κ

2

)|ξ |4

)

= 1

2

((2κ + γ |ξ |2) +

(ν + κ

2

)|ξ |2

)

− 1

2

√((2κ + γ |ξ |2) −

(ν + κ

2

)|ξ |2

)2

+ 4κ2|ξ |2,

λ3 = 1

2

((2κ + γ |ξ |2) +

(ν + κ

2

)|ξ |2

)

+ 1

2

√((2κ + γ |ξ |2) −

(ν + κ

2

)|ξ |2

)2

+ 4κ2|ξ |2,and their corresponding eigenvectors

e1 = (ξ1, ξ2,0),

e2 =(

κiξ2,−κiξ1,

(ν + κ

2

)|ξ |2 − λ2

),

e3 =(

κiξ2,−κiξ1,

(ν + κ

2

)|ξ |2 − λ3

).

Setting

el = el/|el |, for l = 1,2,3,


and

E =

⎛⎜⎜⎝

e1

e2

e3

⎞⎟⎟⎠

T

with T the transpose transform, we have the spectral decomposition

ΛE = EG, for G =⎛⎝λ1 0 0

0 λ2 0

0 0 λ3

⎞⎠ ,

and the isometric property

EET = I or |Eu| = |u|,since A is an self-conjugate operator.

Letting σ1 = 1/2 min{ν, γ }, σ2 = 2(max{ν, γ } + κ/2), we have the crucial relations

σ1|ξ |2 < λl < σ2|ξ |2, for l = 1,2,3.

Therefore, we have∣∣e−Λt u0∣∣ = ∣∣Ee−Gt ET u0

∣∣ = ∣∣e−Gt ET u0∣∣

and

e−σ2|ξ |2t |u0| = e−σ2|ξ |2t ∣∣ET u0∣∣ �

∣∣e−Gt ET u0∣∣ � e−σ1|ξ |2t ∣∣ET u0

∣∣ = e−σ1|ξ |2t |u0|,and thus (2.2) is derived.

Furthermore, by applying Plancherel’s theorem and (2.2), we have∥∥∇me−Atu0∥∥2 = ∥∥ ∇me−Atu0

∥∥2 =∫R2

|ξ |2m∣∣ e−Atu0

∣∣2dξ

� c

∫R2

|ξ |2me−2σ1|ξ |2t |u0|2 dξ � c‖u0‖21

∫R2

|ξ |2me−2σ1|ξ |2t dξ

� ct−m−1‖u0‖21

∞∫0

sme−s ds = ct−m−1‖u0‖21.

This implies (2.3). Similarly, we have∥∥∇e−Atu0∥∥ � ct−

12 ‖u0‖.

In order to show the validity of (2.4), we use the self-conjugate property of the operator A toobtain, for two vector fields u0 and u′

0,∣∣∣∣∫R2

(e−tAu0

) · u′0 dx

∣∣∣∣ =∣∣∣∣∫R2

u0 · (e−tAu′0

)dx

∣∣∣∣� ‖u0‖ 4

3

∥∥e−tAu′0

∥∥4 � ‖u0‖ 4

3

∥∥e−tAu′0

∥∥ 12∥∥∇e−tAu′

0

∥∥ 122

� t−14 ‖u0‖ 4

∥∥u′0

∥∥.
3


This implies∥∥e−tAu0∥∥ � ct−

14 ‖u0‖L 4

3

and so the validity of (2.4) due to (2.3) and the semigroup property of e−tA.The proof of Lemma 2.1 is complete. �With the estimates of semigroup desired in Lemma 2.1, we now begin to investigate the upper

and lower bounds of decay of rates to the linearized equations (1.4)–(1.5).

Lemma 2.2. Suppose that e−Atu0 is a solution of (1.4)–(1.5) with initial vector field u0 ∈ Hm

(m � 0) satisfying (1.6) for γ > 1. Then there exist two positive constants c and c1 such that

c(1 + t)−m+γ

2 �∥∥∇me−Atu0

∥∥ � c1(1 + t)−m+γ

2 , for large t. (2.5)

Proof. The main idea of the proof is borrowed from Oliver and Titi [23]. Applying Plancherel’stheorem and (1.6), (2.2), one shows that

∥∥∇me−Atu0∥∥2 = ∥∥ ∇me−Atu0

∥∥2 =∫R2


∣∣2dξ

� c

∫R2

|ξ |2me−2σ1|ξ |2t ∣∣u0(ξ)∣∣2

dξ

� c

∞∫0

2π∫0

r2m+1e−2σ1r2t

∣∣u0(rθ)∣∣2

dθ dr

� c

t− 1

4∫0

r2m+1e−2σ1r2t ρ(r) dr + ce−2σ1t

12

∞∫t− 1

4

2π∫0

r2m+1∣∣u0(rθ)

∣∣2dθ dr

� ct−m−γ

2σ1t− 1

2∫0

sm+γ−1e−s

[1 +

(o(s/t)

s/t

)γ−1]ds + ce−2σ1t

12∥∥∇mu0

∥∥2,

for large t . The function in between the square brackets is bounded and independent of t . Thisimplies the upper bounds in (2.5). The lower bounds are derived from the following derivation:

∥∥∇me−Atu0∥∥2 = ∥∥ ∇me−Atu0

∥∥2 =∫R2


∣∣2dξ

� c

∫R2

|ξ |2m∣∣e−2σ2|ξ |2t ∣∣u0(ξ)

∣∣∣∣2dξ

= c

∞∫r2m+1e−2σ2r

2t ρ(r) dr

0


� ct−m−γ

1∫0

sm+γ−1e−s ds + o(t−m−γ

)� ct−m−γ , for large t.

Thus the proof of Lemma 2.2 is complete. �3. Auxiliary decay estimates

In this section we present some auxiliary decay estimates that will be used in the followingsections.

Lemma 3.1. Under the same conditions stated in Theorem 1.1, we have the following auxiliarydecay estimates:

t∥∥∇u(t)

∥∥2 → 0, t → ∞, (3.1)∥∥u(t)∥∥ � c(1 + t)−μ, 0 < μ < 1 − γ /2, for large t. (3.2)

Proof. It should be mentioned that the smooth solutions of the two-dimensional micropolar fluidequations (1.1)–(1.2) are globally defined under the smooth initial data [11,19,24]. Therefore wetake L2-inner product of (1.1) with the vector fields (v,w) and integrate by parts to get

d

dt

∥∥u(t)∥∥2 � d

dt

(∥∥v(t)∥∥2 + ∥∥w(t)

∥∥2)= 2

(−

(ν + κ

2

)‖∇v‖2 − γ ‖∇w‖2 − 2κ‖w‖2 + 2κ

∫R2

w∇ × v dx

)

� 2

(−

(ν + κ

2

)‖∇v‖2 − γ ‖∇w‖2 − 2κ‖w‖2 + 2κ‖w‖‖∇v‖

)� −4σ1‖∇u‖2 (by Young inequality). (3.3)

Hence from (3.3), it follows that∥∥u(t)∥∥ �

∥∥u(s)∥∥ and (3.4)

t∫s

∥∥∇u(τ)∥∥2

dτ � 1

4σ1

∥∥u(s)∥∥2

, for t � s � 0. (3.5)

Similarly, by taking the L2-inner product of (1.1) with (�v,�w), one shows that

1

2

d

dt

∥∥∇u(t)∥∥2

� 1

2

d

dt

(∥∥∇v(t)∥∥2 + ∥∥∇w(t)

∥∥2)= −

(ν + κ

2

)‖�v‖2 − γ ‖�w‖2 − 2κ‖∇w‖2 − 2κ

∫R2

(�v) · ∇w dx

+∫

2

(v · ∇v) · �v dx +∫

2

(v · ∇w) · �w dx

R R


� −(

ν + κ

2

)‖�v‖2 − γ ‖�w‖2 − 2κ‖∇w‖2 + 2κ‖�v‖‖∇w‖

+ ‖u‖∞‖∇u‖‖�u‖ (by Hölder inequality)

� −(

ν + κ

2

)‖�v‖2 − γ ‖�w‖2 − 2κ‖∇w‖2 + κ

2‖�v‖2 + 2κ‖∇w‖2

+ ‖u‖ 12 ‖∇u‖‖�u‖ 3

2 (by Gagliardo–Nirenberg inequality and Young inequality)

� −2σ1‖�u‖2 + c‖u0‖ 12 ‖∇u‖‖�u‖ 3

2 � c‖∇u‖4 (by (3.4) and Young inequality

).

This gives

ddt

‖∇u‖2

‖∇u‖2� c‖∇u‖2. (3.6)

Integrating in time from s to t , for t � s � 0, we have

∥∥∇u(t)∥∥2 � exp

{c

t∫s

∥∥∇u(τ)∥∥2

dτ

}∥∥∇u(s)∥∥2

� exp

{c

t∫0

∥∥∇u(τ)∥∥2

dτ

}∥∥∇u(s)∥∥2

� ec

4σ1‖u0‖2∥∥∇u(s)

∥∥2 = c0∥∥∇u(s)

∥∥2, (3.7)

and hence

t

2

∥∥∇u(t)∥∥2 � c0

t∫t2

∥∥∇u(s)w‖2 ds → 0 as t → ∞,

which gives the validity of (3.1).To prove (3.2), we now consider the integral equation formulation of (1.3)

u(t) = e−Atu0 +t∫

0

e−A(t−s)P (v · ∇u)ds. (3.8)

Observing that

ϕ(t) ≡ t12∥∥∇u(t)

∥∥ → 0 (t → ∞) (3.9)

due to (3.1) and then applying (2.3), (2.5) and (3.9) to (3.8), we have

∥∥u(t)∥∥ =

∥∥∥∥∥e−Atu0 +t∫

0

e−A(t−s)P (v · ∇u)ds

∥∥∥∥∥�

∥∥e−Atu0∥∥ +

t∫ ∥∥Pe−A(t−s)(v · ∇u)∥∥ds

0


� c(1 + t)−γ2 + c

t∫0

(t − s)−12 ‖v · ∇u‖1 ds

� c(1 + t)−γ2 + c

t∫0

(t − s)−12 ‖∇u‖‖u‖ds

� c(1 + t)−γ2 + c

t∫0

(t − s)−12 (1 + s)−

12 ϕ(s)

∥∥u(s)∥∥ds. (3.10)

Since 0 < μ < 1 − γ /2, one shows that

(1 + t)μ∥∥u(t)

∥∥ � c + cJ (t) sup0�s�t

(1 + s)μ∥∥u(s)

∥∥, (3.11)

where

J (t) = (1 + t)μ

t∫0

(t − s)−12 (1 + s)−

12 −μϕ(s) ds.

Thus (3.2) holds true whenever cJ (t) < 1/2, for large t , or

J (t) → 0 as t → ∞.

Indeed, for a given small ε > 0, we choose a constant T > 0 so that ϕ(t) < ε (t > T ). We thushave, for t > T ,

J (t) = (1 + t)μ

{ T∫0

+t∫

T

}(t − s)−

12 (1 + s)−

12 −μϕ(s) ds

� c(1 + t)μ(t − T )−12

T∫0

(1 + s)−12 −μ ds

+ ε(1 + t)μ

t∫T

(t − s)−12 (1 + s)−

12 −μ ds

� ct−( 12 −μ) + cε → cε as t → ∞.

This completes the proof of Lemma 3.1. �4. Derivation on the upper bounds

Considering the mild formulation of higher order derivative of (1.1)

∇mu(t) = ∇me−Atu0 +t∫

0

∇me−A(t−s)P (v · ∇u)ds, (4.1)

and then applying (2.3)–(2.5), (3.9) to (4.1), one shows that


∥∥∇mu(t)∥∥ = ∥∥∇me−Atu0‖ +

t∫0

∥∥∇me−A(t−s)P (v · ∇u)∥∥ds

= ∥∥∇me−Atu0∥∥ +

t∫0

∥∥P∇me−A(t−s)(v · ∇u)∥∥ds

�∥∥∇me−Atu0

∥∥ +2∑

i=1

t2∫

0

∥∥∇m+1e−A(t−s)(viu)∥∥ds

+2∑

i=1

t∫t2

(t − s)−34∥∥∇m(viu)

∥∥L

43ds

� c(1 + t)−m+γ

2 + c

t2∫

0

(t − s)−m+2

2 ‖u‖2 ds

+ c

t∫t2

(t − s)−34 ‖u‖ 1

2 ‖∇u‖ 12∥∥∇mu

∥∥ds

� c(1 + t)−m+γ

2 + c

t2∫

0

(t − s)−m+2

2 (1 + s)−μ‖u‖ds

+ c

t∫t2

(t − s)−34 (1 + s)−

1+2μ4

∥∥∇mu∥∥ds. (4.2)

Here we have used the inequality

∥∥∇m(viu)∥∥ 4

3� 2‖u‖4

∥∥∇mu∥∥ +

m−1∑k=1

∥∥∇m−ku∥∥

4

∥∥∇ku∥∥

� c‖u‖ 12 ‖∇u‖ 1

2∥∥∇mu

∥∥ (4.3)

due to the Hölder inequality and the Gagliardo–Nirenberg inequality (see, for example, [29]).When m = 0, it follows from (4.2) that

(1 + t)γ2∥∥u(t)

∥∥ � c + J0(t) sup0�s�t

(1 + s)γ2∥∥u(s)

∥∥ � c + 1

2sup

0�s�t

(1 + s)γ2∥∥u(s)

∥∥,

for t sufficiently large. Here the function

J0(t) = c(1 + t)γ2

t2∫(t − s)−1(1 + s)−μ− γ

2 ds

0


+ c(1 + t)γ2

t∫t2

(t − s)−34 (1 + s)−

14 −γ ds → 0 (4.4)

as t → ∞. Thus the estimate∥∥u(t)∥∥ � c(1 + t)−

γ2 (4.5)

holds true.Similarly, the upper bound with m > 0 follows from the estimate

(1 + t)m+γ

2∥∥∇mu(t)

∥∥ � c + Jm(t) sup0�s�t

(1 + s)m+γ

2∥∥∇mu(s)

∥∥due to (4.2) and the property Jm(t) → 0 as t → ∞, where

Jm(t) = c(1 + t)m+γ

2

t2∫

0

(t − s)−m+2

2 (1 + s)−μ− γ2 ds

+ c(1 + t)m+γ

2

t∫t2

(t − s)−34 (1 + s)−

1+2μ4 − m+γ

2 ds

� ctγ2 −1

t2∫

0

(1 + s)−μ− γ2 ds + c(1 + t)−

1+2μ4

t∫t2

(t − s)−34 ds, (4.6)

for t > 1. Thus we have the upper bounds decay estimate as stated in Theorem 2.1,∥∥∇mu(t)∥∥ � c(1 + t)−

m+γ2 , for large t and m � 0. (4.7)

5. Derivation on the lower bounds

Considering the error estimates of solutions between the micropolar fluid equations (1.3) andthe linearized equations (1.4), we have from (4.1) that∥∥∇mu(t) − ∇me−Atu0

∥∥=

∥∥∥∥∥t∫

0

∇me−A(t−s)P (v · ∇u)ds

∥∥∥∥∥�

t∫0

∥∥P∇me−A(t−s)(v · ∇u)∥∥ds

�2∑

i=1

t2∫

0

∥∥∇m+1e−A(t−s)(viu)∥∥ds +

2∑i=1

t∫t

(t − s)−34∥∥∇m(viu)

∥∥L

43ds

2


� c

t2∫

0

(t − s)−m+2

2 ‖u‖2 ds + c

t∫t2

(t − s)−34 ‖u‖ 1

2 ‖∇u‖ 12∥∥∇mu

∥∥ds

� c

t2∫

0

(t − s)−m+2

2 (1 + s)−μ− γ2 ds + c

t∫t2

(t − s)−34 (1 + s)−

m+γ2 − 1+2γ

4 ds

� c(1 + t)−m+γ

2 −μ + c(1 + t)−m+γ

2 − γ2 , for large t, (5.1)

where the use is made of (2.3), (2.4), (3.2), (4.3), (4.7).Hence, by (2.5), we have∥∥∇mu(t)

∥∥ �∥∥∇me−Atu0

∥∥ − ∥∥∇mu(t) − ∇me−Atu0∥∥

� c(1 + t)−m+γ

2 , for large t, (5.2)

which proves the lower bounds on the decay of ‖∇mu(t)‖ as stated in Theorem 1.1. The proofof Theorem 1.1 is complete. �Acknowledgment

The authors would like to express their gratitude to the referee for his/her valuable comments and suggestions.

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On upper and lower bounds of higher order derivatives for solutions to the 2D micropolar fluid equations