Time Decay Rates of the Micropolar Equations with Zero Angular Viscosity

Abstract

This paper is concerned with the large time decay rates of the two-dimensional (2D) micropolar equations with zero angular viscosity. Based on the generalized Fourier splitting methods and low frequency effect analysis, we firstly obtain the solutions decay as $\Vert u\Vert _{L^2}+ \Vert \omega \Vert _{L^2}\le C(1+t)^{-\frac{1}{2}}$. Moreover, by exploring the new structure of the system, we obtain a new improved decay rates $\Vert \omega \Vert _{L^2}+ \Vert \nabla u\Vert _{L^2}\le C(1+t)^{-1}$. Our methods here are also available to the time decay issue of the complex fluid flows with partial dissipation.

1 Introduction

Consider the two-dimensional (2D) incompressible micropolar fluid flows

$$\begin{aligned} \begin{aligned}&\partial _{t}u+(u\cdot \nabla )u+\nabla \pi =(\mu +\chi )\Delta u+2\chi \nabla \times \omega ,\\&\partial _{t}\omega +(u\cdot \nabla )\omega +4\chi \omega =\gamma \Delta \omega +2\chi \nabla \times u,\\&\nabla \cdot u=0,\\&u(x, 0)=u_{0}(x),\quad \omega (x,0)=\omega _{0}(x), \end{aligned} \end{aligned}$$

(1.1)

where $u(x,y,t)=(u_{1}(x,y,t),\,u_{2}(x,y,t))$ is the unknown velocity vector field, $\pi (x,y,t)$ and $\omega (x,y,t)$ are the unknown scalar pressure field and unknown scalar micro-rotation angular velocity of the rotation of the particles of the fluid, respectively. $\mu \ge 0$ is the Newtonian kinetic viscosity and $\chi \ge 0$ is the dynamics micro-rotation viscosity, and $\gamma \ge 0$ is the angular viscosity. It is worth noting that

$$\begin{aligned} \nabla \times u=\partial _{x_{1}}u_{2}-\partial _{x_{2}}u_{1}, \quad \nabla \times \omega =(\partial _{x_{2}}\omega ,\, -\partial _{x_{1}}\omega ). \end{aligned}$$

The micropolar equations introduced by Eringen [5] describe the motion of numerous real fluids with the micro-structure and nonsymmetric stress tensor(eg, blood). When scalar micro-rotation angular velocity field $\omega$ is neglected, the micropolar fluid motion model (1.1) reduces to the Navier–Stokes equations [12].

Because of their mathematically significant, the well-posedness and large time behavior of the micropolar equations attract much attention and many important results have been achieved. For example, we can refer to the global existence and uniqueness for 2D or 3D micropolar equations with full viscosity (namely, $\mu>0,\, \chi >0$ and $\gamma >0$) [3, 4, 6,7,8,9,10, 13] for the partial dissipation. For the large time behavior, Dong and Chen [2] derived large time $L^{2}$ decay rates of solutions to the 2D micropolar fluid flows with full viscosity. When (1.1) involves only the angular viscosity dissipation, Dong et al. [4]recently obtained the large time behavior of the global regular solutions which is based on a diagonalization process to eliminate the linear terms together with an auxiliary decay of $\nabla u,\nabla \omega$.

Motivated by the decay results of micropolar flows [2, 4] and related fluid models [16,17,18], the main purpose of this paper is to investigate the upper bounds of time decay rates of global solution to the 2D micropolar equations with only velocity dissipation and zero angular viscosity,

$$\begin{aligned} \begin{aligned}&\partial _{t}u+(u\cdot \nabla )u=(\mu +\chi )\Delta u-\nabla \pi +2\chi \nabla \times \omega ,\\&\partial _{t}\omega +(u\cdot \nabla )\omega +4\chi \omega =2\chi \nabla \times u,\\&\nabla \cdot u=0,\\&u(x, 0)=u_{0}(x),\quad \omega (x,0)=\omega _{0}(x). \end{aligned} \end{aligned}$$

(1.2)

More precisely, our first result is the based energy decay of the global weak solutions.

Theorem 1.1

Let $(u,\, \omega )$ be the global weak solution of the 2D micropolar equations (1.2) with $(u_0,\, \omega _0)\in L^1(\mathbb {R}^2)\cap L^2(\mathbb {R}^2)$, then the solution $(u,\, \omega )$ has the following upper decay rates

$$\begin{aligned} \Vert u\Vert _{L^2}+\Vert \omega \Vert _{L^2}\le C(1+t)^{-\frac{1}{2}}. \end{aligned}$$

Remark 1.2

On comparison with the optimal decay rates of 2D Navier-Stokes equations, here $\omega _0$ is not divergence free, it is an interesting issue to improve the decay rate of $\Vert u\Vert _{L^{2}(\mathbb {R}^2)}$. We will consider this problem in future.

Due to the special structure of the 2D micropolar equations (1.2), we also can improved the above decay rates for regular solution.

Theorem 1.3

Let $u_0\in L^1(\mathbb {R}^2)\cap H^1(\mathbb {R}^2)$ and $\omega _0\in L^1(\mathbb {R}^2)\cap L^2(\mathbb {R}^2)$ such that $\nabla \cdot u_{0}=0$. Let $(u,\, \omega )$ be the global solution of the system (1.2) and $\mu =\chi ,$ then we have the following improved decay rates

$$\begin{aligned} \Vert \omega \Vert _{L^2}+\Vert \nabla u\Vert _{L^2}\le C(1+t)^{-1}. \end{aligned}$$

Remark 1.4

According to the results in Dong and Zhang [3], the bounds of higher derivatives for the global solutions are exponential in time,

$$\begin{aligned} \Vert (-\Delta )^m u\Vert _{L^2}\le Ce^{Ct},\ \ m\ge 1, \end{aligned}$$

it is difficult to establish the time decay estimates for the second order derivatives or higher derivatives for the global solutions.

On comparison with the previous results [1, 4], in order to prove the decay results of Theorem 1.3, the two main difficulties should be come over. Firstly we can not obtain directly the upper bound estimates of Fourier transformation of the solution $(u,\omega )$ by the classic Fourier splitting methods (refer to [11, 14, 15]). The main obstacle lies in the lack of full dissipation on micro-rotation angular velocity. In order to explore the low frequency effect of the global solution, we consider the summation of equations and their conjugate formation and fortunately find the system still obeys a good structure

$$\begin{aligned} |\hat{u}|+|\hat{\omega }|\le C +C|\xi |\int ^{t}_{0}\left( \Vert u\Vert ^2_{L^2}+\Vert \omega \Vert ^2_{L^2}\right) \hbox {d}\tau , \quad |\xi |<1. \end{aligned}$$

This observation allows us to develop the generalized Fourier splitting method from auxiliary logarithmic decay to the algebraic decay, i.e.

$$\begin{aligned} \Vert u\Vert _{L^2}+\Vert \omega \Vert _{L^2}\le C(1+t)^{-\frac{1}{2}}. \end{aligned}$$

In order to examine the improved decay rates of $\omega , \nabla u$, the second difficulty is that it seems difficult to borrow the idea of Dong et al. [4] by applying diagonalization process. In our case here, indeed we have not the auxiliary decay of $\nabla u,\nabla \omega$ stated in [4] and the similar linear $L^p-L^q$ estimates of heat semigroup. Thus the complex diagonalization process in [4] is not valid for the improved decay rates of $\omega , \nabla u$. It should mentioned that in the study of the global regularity of the system (1.2), Dong and Zhang [3] introduced a new function $Z=\Omega -\frac{2\chi }{\mu +\chi }\omega \ ( \Omega =\nabla \times u)$ which satisfies the following equation

$$\begin{aligned} \partial _{t}Z+(u\cdot \nabla )Z-(\mu +\chi )\Delta Z + \frac{4\chi ^{2}}{\mu +\chi } Z =\left( \frac{8\chi ^{2}}{\mu +\chi }-\frac{8\chi ^{3}}{(\mu +\chi )^{2}}\right) \omega . \end{aligned}$$

(1.3)

We observe that that equation is also helpful for the time decay issue. More precisely, under the energy estimates, we may check

$$\begin{aligned} \Vert Z(t)\Vert ^{2}_{L^{2}}+\Vert \omega (t)\Vert ^{2}_{L^{2}}\le C\left( \Vert Z(s)\Vert ^{2}_{L^{2}}+\Vert \omega (s)\Vert ^{2}_{L^{2}}\right) , \ \ 0\le s<t\le \infty . \end{aligned}$$

Moreover, the special structure of the system (1.2) allows us to derive an auxiliary decay of $\Vert \nabla u(s)\Vert ^{2}_{L^{2}}, \Vert \omega (s)\Vert ^{2}_{L^{2}}$

$$\begin{aligned} \int _{0}^{t}(1+s)^{n}(\Vert \nabla u(s)\Vert ^{2}_{L^{2}}+\Vert \omega (s)\Vert ^{2}_{L^{2}})\hbox {d}s \le C(1+t)^{n-1},\qquad \hbox {for}\quad n\gg 1. \end{aligned}$$

Thus it opens a window for us to derive the improved decay rates of $\nabla u,\omega$ (see the next section for details). Indeed, we will show that

$$\begin{aligned} \Vert \nabla u\Vert _{L^{2}}+\Vert \omega (t)\Vert _{L^{2}} \le C(1+t)^{-1}. \end{aligned}$$

The rest of this paper is divided into three sections. In the second section we explore the low frequency effect of the 2D micropolar equations (1.2). By developing the generalized Fourier splitting methods, we proved Theorem 1.1 for the energy decay of $u,\omega$ in Sect. 3. In Sect. 4, we improved the decay rates for $\nabla u,\omega$ by applying some new observation and difference analysis methods. It should be mentioned that the technique present in Sects. 3 and 4 can be applied widely to some revelent fluid models with partial dissipation.

2 Low-Frequency Effect of Solutions

Since the 2D micropolar equations (1.2) have the difference dissipative mechanism with $\Delta u$ and linear damping $\omega$. The classic Schonbek’s Fourier splitting methods can not apply directly. The main obstacle lies in lack of the low frequency effect of the system. The Kato’s methods which is based on the $L^p-L^q$ estimates of heat semigroup is also not valid due to the limitation of linear damping $\omega .$ In order to come over the main difficulty, we observe that the summation of the system in frequency and their conjugate form has an elegant structure, which allows us to explore the low frequency effect of the solutions.

Proposition 2.1

Let $(u,\, \omega )$ be the global solutions of the equations (1.2) with $(u_0,\, \omega _0)\in L^1(\mathbb {R}^2)\cap L^2(\mathbb {R}^2)$, one has the following low frequency effect for $(u,\, \omega )$

$$\begin{aligned} |\hat{u}|+|\hat{\omega }|\le C +C|\xi |\int ^{t}_{0}\left( \Vert u\Vert ^2_{L^2}+\Vert \omega \Vert ^2_{L^2}\right) \hbox {d}\tau , \quad |\xi |^2\le 1. \end{aligned}$$

(2.1)

Proof

Taking the $L^{2}$-inner product of (1.2) with u and $\omega$, respectively, and then adding the resulting equations together, it yields

$$\begin{aligned}&\frac{1}{2}\frac{\hbox {d}}{\hbox {d}t}\left( \Vert u\Vert ^{2}_{L^{2}}+\Vert \omega \Vert ^{2}_{L^{2}}\right) +(\mu +\chi )\Vert \nabla u\Vert ^{2}_{L^{2}}+4\chi \Vert \omega \Vert ^{2}_{L^{2}}\\&\quad =2\chi \int _{\mathbb {R}^{2}}\nabla \times \omega \cdot u \hbox {d}x\hbox {d}y+2\chi \int _{\mathbb {R}^{2}}(\nabla \times u)\omega \hbox {d}x\hbox {d}y\\&\quad =4\chi \int _{\mathbb {R}^{2}}(\nabla \times u)\omega \hbox {d}x\hbox {d}y\\&\quad \le 4\chi \Vert \nabla u\Vert _{L^{2}}\Vert \omega \Vert _{L^{2}}\\&\quad \le \frac{\mu +2\chi }{2}\Vert \nabla u\Vert ^{2}_{L^{2}}+\frac{4\chi ^{2}}{\mu +2\chi }\Vert \omega \Vert ^{2}_{L^{2}}, \end{aligned}$$

which obeys

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t}\left( \Vert u\Vert ^{2}_{L^{2}}+\Vert \omega \Vert ^{2}_{L^{2}}\right) +\mu \Vert \nabla u\Vert ^{2}_{L^{2}}+\frac{8\mu \chi }{\mu +2\chi }\Vert \omega \Vert ^{2}_{L^{2}} \le 0. \end{aligned}$$

(2.2)

Integrating in time, we obtain the following basic energy inequality

$$\begin{aligned}&\Vert u(t)\Vert ^{2}_{L^{2}}+\Vert \omega (t)\Vert ^{2}_{L^{2}} +\mu \int ^{t}_{0}\Vert \nabla u(\tau )\Vert ^{2}_{L^{2}}\hbox {d}\tau +\frac{8\mu \chi }{\mu +2\chi }\int ^{t}_{0} \Vert \omega (\tau )\Vert ^{2}_{L^{2}}\hbox {d}\tau \nonumber \\&\quad \le \Vert u_{0}\Vert ^{2}_{L^{2}}+\Vert \omega _{0}\Vert ^{2}_{L^{2}}. \end{aligned}$$

(2.3)

Taking the Fourier Transform to the micropolar equations (1.2) obeys

$$\begin{aligned} \begin{aligned}&\partial _{t}\hat{u}+(\mu +\chi )|\xi |^{2}\hat{u}=\mathcal {F} [-\nabla \pi -(u\cdot \nabla )u+2\chi \nabla \times \omega ],\\&\partial _{t}\hat{\omega }+4\chi \hat{\omega }=\mathcal {F} [-(u\cdot \nabla )\omega +2\chi \nabla \times u]. \end{aligned} \end{aligned}$$

Multiplying the above first and second equations by their conjugate functions $\bar{\hat{u}}$ and $\bar{\hat{\omega }}$, respectively, and summing the resulting equations, we have after taking the real part and noting that

$$\begin{aligned} |\hat{u}|^{2}=\hat{u}\cdot \bar{\hat{u}}, \end{aligned}$$

$$\begin{aligned}&\frac{1}{2}\frac{\partial }{\partial t}(|\hat{u}|^{2}+|\hat{\omega }|^{2}) +(\mu +\chi )|\xi |^2|\hat{u}|^{2}+4\chi |\hat{\omega }|^{2}\nonumber \\&\quad =\mathcal {R}e \left\{ \mathcal {F}[-\nabla \pi ]\cdot \bar{\hat{u}}+\mathcal {F}[-(u\cdot \nabla )u] \cdot \bar{\hat{u}}+\mathcal {F}[2\chi \nabla \times \omega ]\cdot \bar{\hat{u}} \right\} \nonumber \\&\qquad +\mathcal {R}e\left\{ \mathcal {F}[-(u\cdot \nabla )\omega ]\bar{\hat{\omega }}+\mathcal {F} [2\chi \nabla \times u]\bar{\hat{\omega }}\right\} . \end{aligned}$$

(2.4)

In order to estimate the right hand side of (2.4), we firstly take the divergence to the Eq. (1.2)$_{1}$

$$\begin{aligned} \pi= & {} \Delta ^{-1}\nabla \cdot [-u\cdot \nabla u]\\= & {} (-\Delta ^{-1})\nabla \otimes \nabla (u\otimes u), \end{aligned}$$

and

$$\begin{aligned} \mathcal {F}[-\nabla \pi ]\cdot \bar{\hat{u}}\le & {} |\xi ||\hat{\pi }||\bar{\hat{u}}|\\\le & {} |\xi |\frac{|\xi \otimes \xi |}{|\xi |^{2}}|\widehat{u\otimes u}||\hat{u}|\\\le & {} |\xi |\Vert u\otimes u\Vert _{L^1}|\hat{u}|\\\le & {} |\xi |\Vert u\Vert _{L^2}^2|\hat{u}|. \end{aligned}$$

Using divergence free condition, we also have

$$\begin{aligned} \mathcal {F}[-(u\cdot \nabla )u]\cdot \bar{\hat{u}}\le & {} |\xi ||\widehat{u\otimes u}||\hat{u}|\\\le & {} |\xi |\Vert u\otimes u\Vert _{L^1}|\hat{u}|\\\le & {} |\xi |\Vert u\Vert _{L^2}^2|\hat{u}|, \end{aligned}$$

similarly,

$$\begin{aligned} \mathcal {F}[-(u\cdot \nabla )\omega ]\bar{\hat{\omega }}\le & {} |\xi ||\widehat{u\otimes \omega }||\bar{\hat{\omega }}|\\\le & {} |\xi |\Vert u\otimes \omega \Vert _{L^1}|\hat{\omega }|\\\le & {} C|\xi |\left( \Vert u\Vert _{L^2}^2+\Vert \omega \Vert _{L^2}^2\right) |\hat{\omega }|. \end{aligned}$$

Applying Young’s inequality, we obey

$$\begin{aligned} \mathcal {F}[2\chi \nabla \times \omega ]\cdot \bar{\hat{u}}+\mathcal {F} [2\chi \nabla \times u]\bar{\hat{\omega }}\le & {} 4\chi |\xi | |\hat{\omega }||\hat{u}|\\\le & {} \frac{\mu +2\chi }{2}|\xi |^2|\hat{u}|^2+\frac{8\chi ^2}{\mu +2\chi }|\hat{\omega }|^2. \end{aligned}$$

Inserting the above estimates into (2.4) yields

$$\begin{aligned} \frac{1}{2}\frac{\partial }{\partial t}(|\hat{u}|^2+|\hat{\omega }|^2) +\frac{\mu }{2}|\xi |^2|\hat{u}|^{2}+\frac{4\mu \chi }{\mu +2\chi }|\hat{\omega }|^{2} \le C|\xi |\left( \Vert u\Vert ^2_{L^2}+\Vert \omega \Vert ^2_{L^2}\right) (|\hat{u}|+|\hat{\omega }|), \end{aligned}$$

or

$$\begin{aligned}&\frac{\partial }{\partial t}(|\hat{u}|^2+|\hat{\omega }|^2) +2\min \left\{ \frac{\mu }{2},\,\frac{4\mu \chi }{\mu +2\chi }\right\} (|\xi |^2|\hat{u}|^{2} +|\hat{\omega }|^{2})\\&\quad \le C|\xi |\left( \Vert u\Vert ^2_{L^2}+\Vert \omega \Vert ^2_{L^2}\right) (|\hat{u}|+|\hat{\omega }|). \end{aligned}$$

In the sake of simplicity, making $M(t)=\sqrt{|\hat{u}|^2+|\hat{\omega }|^2}$ and $\varepsilon =\min \{\frac{\mu }{2},\,\frac{4\mu \chi }{\mu +2\chi }\}$ and noting that $|\xi |^2\le 1$, we have

$$\begin{aligned} \frac{\partial }{\partial t}[M(t)]^2+2\varepsilon |\xi |^2[M(t)]^2\le C|\xi |\left( \Vert u\Vert ^2_{L^2}+\Vert \omega \Vert ^2_{L^2}\right) M(t), \end{aligned}$$

namely,

$$\begin{aligned} \frac{\partial }{\partial t}M(t)+\varepsilon |\xi |^2M(t) \le C|\xi |\left( \Vert u\Vert ^2_{L^2}+\Vert \omega \Vert ^2_{L^2}\right) . \end{aligned}$$

Integrating in time from 0 to t, it follows that

$$\begin{aligned} M(t)\le & {} e^{-\varepsilon |\xi |^{2}t}M_0+\int ^{t}_{0}C|\xi | \left( \Vert u\Vert ^2_{L^2}+\Vert \omega \Vert ^2_{L^2}\right) e^{- \varepsilon |\xi |^{2}(t-\tau )}\hbox {d}\tau \nonumber \\\le & {} e^{-\varepsilon |\xi |^{2}t}\sqrt{|\hat{u}_0|^2+|\hat{\omega }_0|^2} +C|\xi |\int ^{t}_{0}\left( \Vert u\Vert ^2_{L^2}+\Vert \omega \Vert ^2_{L^2}\right) \hbox {d}\tau \nonumber \\\le & {} C+C|\xi |\int ^{t}_{0}\left( \Vert u\Vert ^2_{L^2}+\Vert \omega \Vert ^2_{L^2}\right) \hbox {d}\tau . \end{aligned}$$

This completes the proof of Proposition 2.1.$\square$

3 Proof of Theorem 1.1

In the section, we apply the generalized Fourier splitting methods to the partially dissipative micropolar equations.

Let

$$\begin{aligned} \mathcal {B}(t)=\left\{ \xi \in \mathbb {R}^{2}\big |\,|\xi |^2\le \frac{f'(t)}{2\varepsilon f(t)}\right\} , \qquad \mathcal {B}(t)^c=\mathbb {R}^2\backslash \mathcal {B}(t), \end{aligned}$$

where $f(t)\in C^{\infty }[0,\,\infty )$ is a positive function with t and satisfies

$$\begin{aligned} f(0)=1,\quad f'(t)>0\quad \hbox {and}\quad \frac{f'(t)}{2\varepsilon f(t)}\le 1,\quad \forall \, t>t_{0}>1. \end{aligned}$$

Multiplying both sides of inequality (2.2) by f(t) yields

$$\begin{aligned}&\frac{\hbox {d}}{\hbox {d}t}\left[ f(t)\left( \Vert u\Vert ^2_{L^2}+\Vert \omega \Vert ^2_{L^2}\right) \right] +2\varepsilon f(t)\left( \Vert \nabla u\Vert ^2_{L^2}+\Vert \omega \Vert ^2_{L^2}\right) \nonumber \\&\quad \le f'(t)\left( \Vert u\Vert ^2_{L^2}+\Vert \omega \Vert ^2_{L^2}\right) , \end{aligned}$$

(3.1)

then using the Plancherel Theorem yields

$$\begin{aligned}&\frac{\hbox {d}}{\hbox {d}t}\left[ f(t)\left( \Vert \hat{u}\Vert ^2_{L^2}+\Vert \hat{\omega }\Vert ^2_{L^2}\right) \right] +2\varepsilon f(t)\int _{\mathbb {R}^{2}}(|\xi |^2|\hat{u}(\xi )|^2+|\hat{\omega }(\xi )|^2) \hbox {d}\xi \nonumber \\&\quad \le f'(t)\int _{\mathbb {R}^{2}}(|\hat{u}(\xi )|^2+|\hat{\omega }(\xi )|^2)\hbox {d}\xi . \end{aligned}$$

(3.2)

Thanks to

$$\begin{aligned}&2\varepsilon f(t)\int _{\mathbb {R}^{2}}(|\xi |^2|\hat{u}(\xi )|^2+|\hat{\omega }(\xi )|^2)\hbox {d}\xi +f'(t)\int _{\mathcal {B}(t)}(|\hat{u}(\xi )|^2+|\hat{\omega }(\xi )|^2)\hbox {d}\xi \nonumber \\&\quad \ge 2\varepsilon f(t)\int _{\mathcal {B}(t)^c}(|\xi |^2(|\hat{u}(\xi )|^2+|\hat{\omega }(\xi )|^2)\hbox {d}\xi +f'(t)\int _{\mathcal {B}(t)}(|\hat{u}(\xi )|^2+|\hat{\omega }(\xi )|^2)\hbox {d}\xi \nonumber \\&\quad \ge f'(t)\int _{\mathcal {B}(t)^c}(|\hat{u}(\xi )|^2+|\hat{\omega }(\xi )|^2)\hbox {d}\xi +f'(t)\int _{\mathcal {B}(t)}(|\hat{u}(\xi )|^2+|\hat{\omega }(\xi )|^2)\hbox {d}\xi \nonumber \\&\quad \ge f'(t)\int _{\mathbb {R}^{2}}(|\hat{u}(\xi )|^2+|\hat{\omega }(\xi )|^2)\hbox {d}\xi , \end{aligned}$$

(3.3)

therefore, we obtain

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t}\left[ f(t)\left( \Vert \hat{u}\Vert ^2_{L^2}+\Vert \hat{\omega }\Vert ^2_{L^2}\right) \right] \le f'(t)\int _{\mathcal {B}(t)}(|\hat{u}(\xi )|^2+|\hat{\omega }(\xi )|^2)\hbox {d}\xi . \end{aligned}$$

(3.4)

Employing Proposition 2.1 to the right hand side term of (3.4), we have

$$\begin{aligned}&\int _{\mathcal {B}(t)}(|\hat{u}(\xi )|^2+|\hat{\omega }(\xi )|^2)\hbox {d}\xi \\&\quad \le C\int _{\mathcal {B}(t)}\left[ 1 +|\xi |\int _0^t\left( \Vert u\Vert ^2_{L^2}+\Vert \omega \Vert ^2_{L^2}\right) \hbox {d}\tau \right] ^2\hbox {d}\xi \\&\quad \le C\frac{f'(t)}{f(t)}+C\frac{[f'(t)]^2}{f^2(t)}\left[ \int _0^t\left( \Vert u\Vert ^2_{L^2} +\Vert \omega \Vert ^2_{L^2}\right) \hbox {d}\tau \right] ^2. \end{aligned}$$

Inserting the above inequality into (3.4) and integrating in time obey

$$\begin{aligned}&f(t)\left( \Vert \hat{u}\Vert ^2_{L^2}+\Vert \hat{\omega }\Vert ^2_{L^2}\right) \nonumber \\&\quad \le \left( \Vert u_0\Vert ^2_{L^2}+\Vert \omega _0\Vert ^2_{L^2}\right) +C\int ^t_0\frac{[f'(s)]^2}{f(s)}\hbox {d}s\nonumber \\&\qquad +C\int ^t_0\frac{[f'(s)]^3}{f^2(s)}\left[ \int _0^s\left( \Vert u\Vert ^2_{L^2} +\Vert \omega \Vert ^2_{L^2}\right) \hbox {d}\tau \right] ^2\hbox {d}s. \end{aligned}$$

(3.5)

Now choosing $f(t)=[\ln (e+t)]^3$, then we have the fact

$$\begin{aligned} \frac{[f'(t)]^2}{f(t)}=\frac{C\ln (e+t)}{(e+t)^2},\quad \frac{[f'(t)]^3}{f^2(t)}=\, \frac{C}{(e+t)^3}. \end{aligned}$$

(3.6)

Inserting (3.6) into (3.5), and then applying (2.2), we get

$$\begin{aligned}&[\ln (e+t)]^3\left( \Vert \hat{u}\Vert ^2_{L^2}+\Vert \hat{\omega }\Vert ^2_{L^2}\right) \nonumber \\&\quad \le C+C\int ^t_0\frac{\ln (e+s)}{(e+s)^{2}}\hbox {d}s +C\int ^t_0\frac{1}{(e+s)^3}\left[ \int _0^s\left( \Vert u\Vert ^2_{L^2} +\Vert \omega \Vert ^2_{L^2}\right) \hbox {d}\tau \right] ^2\hbox {d}s\nonumber \\&\quad \le C+C\ln (e+t), \end{aligned}$$

(3.7)

therefore, applying Plancherel Theorem, we have the auxiliary logarithmic decay of solutions

$$\begin{aligned} \Vert u\Vert ^2_{L^2}+\Vert \omega \Vert ^2_{L^2}\le & {} C\,[\ln (e+t)]^{-3}+C\,[\ln (e+t)]^{-2}\\\le & {} C\,[\ln (e+t)]^{-2}. \end{aligned}$$

Now we choose another $f(t) = (1 + t)^2$ and insert it into (3.5) together with Hölder inequality,

$$\begin{aligned}&(1+t)^2\left( \Vert \hat{u}\Vert ^2_{L^2}+\Vert \hat{\omega }\Vert ^2_{L^2}\right) \\&\quad \le \left( \Vert u_0\Vert ^2_{L^2}+\Vert \omega _0\Vert ^2_{L^2}\right) +C\int ^t_0\hbox {d}s +C\int ^t_0\frac{1}{1+s}\left[ \int _0^s\left( \Vert u\Vert ^2_{L^2} +\Vert \omega \Vert ^2_{L^2}\right) \hbox {d}\tau \right] ^2\hbox {d}s\\&\quad \le C+C(1+t)+C\int ^t_0\frac{s}{1+s}\int _0^s\left( \Vert u\Vert ^4_{L^2} +\Vert \omega \Vert ^4_{L^2}\right) \hbox {d}\tau \hbox {d}s\\&\quad \le C(1+t)+C(1+t)\int _0^t\left( \Vert u\Vert ^4_{L^2} +\Vert \omega \Vert ^4_{L^2}\right) \hbox {d}s\\&\quad \le C(1+t)+C(1+t)\int _0^t[\ln (e+s)]^{-2}\left( \Vert u\Vert ^2_{L^2} +\Vert \omega \Vert ^2_{L^2}\right) \hbox {d}s. \end{aligned}$$

Then we obtain

$$\begin{aligned}&(1+t)\left( \Vert u\Vert ^2_{L^2}+\Vert \omega \Vert ^2_{L^2}\right) \nonumber \\&\quad \le C+C\int _0^t(1+s)^{-1}[\hbox {ln}(e+s)]^{-2}(1+s)\left( \Vert u\Vert ^2_{L^2} +\Vert \omega \Vert ^2_{L^2}\right) \hbox {d}s. \end{aligned}$$

(3.8)

Let

$$\begin{aligned} \Phi (t)= & {} (1+t)\left( \Vert u\Vert ^2_{L^2}+\Vert \omega \Vert ^2_{L^2}\right) ,\\ h(t)= & {} (1+t)^{-1}[\ln (e+t)]^{-2}, \end{aligned}$$

then (3.8) becomes

$$\begin{aligned} \Phi (t)\le C+C\int _0^{t}h(s)\Phi (s)\hbox {d}s. \end{aligned}$$

Thanks to

$$\begin{aligned} \int ^\infty _{0}h(s)\hbox {d}s= & {} \int ^\infty _{0}(1+s)^{-1}[\ln (e+s)]^{-2}\hbox {d}s <\infty , \end{aligned}$$

applying the Gronwall inequality to (3.8) yields

$$\begin{aligned} \Phi (t)\le \Phi (0)e^{C\int ^t_{0}h(s)\hbox {d}s}\le C, \end{aligned}$$

which implies the decay

$$\begin{aligned} \Vert u\Vert _{L^2}+\Vert \omega \Vert _{L^2}\le C(1+t)^{-\frac{1}{2}}. \end{aligned}$$

Thus the proof of Theorem 1.1 is completed.$\square$

4 The Proof of Theorem 1.3

In this section, we will show the improved decay rates of global regular solutions for the 2D micropolar equations (1.2).

In the study of the global regularity of the system (1.2), Dong and Zhang introduced a new function $Z=\Omega -\frac{2\chi }{\mu +\chi }\omega \ ( \Omega =\nabla \times u)$ which plays an important role in [3]. We observe the new function is also helpful for the large time decay issue. Indeed, taking the $\nabla \times$ to the first equation of (1.2)$_{1}$ and subtracting $\frac{2\chi }{\mu +\chi }\times (1.2)_{2}$ yields

$$\begin{aligned} \partial _{t}Z+(u\cdot \nabla )Z-(\mu +\chi )\Delta Z + \frac{4\chi ^{2}}{\mu +\chi } Z=\left( \frac{8\chi ^{2}}{\mu +\chi }-\frac{8\chi ^{3}}{(\mu +\chi )^{2}}\right) \omega . \end{aligned}$$

(4.1)

Taking the $L^{2}$-inner products of the Eqs. (4.1) and (1.2)$_{2}$ with Z and $\omega$, respectively, we have

$$\begin{aligned} \frac{1}{2}\frac{\hbox {d}}{\hbox {d}t}\Vert Z\Vert ^{2}_{L^{2}} +\frac{4\chi ^{2}}{\mu +\chi }\Vert Z\Vert ^{2}_{L^{2}}\le & {} \left( \frac{8\chi ^{2}}{\mu +\chi }-\frac{8\chi ^{3}}{(\mu +\chi )^{2}}\right) \int _{\mathbb {R}^{2}}\omega Z \hbox {d}x\hbox {d}y,\\\le & {} \left( \frac{8\chi ^{2}}{\mu +\chi }-\frac{8\chi ^{3}}{(\mu +\chi )^{2}}\right) \Vert Z\Vert _{L^{2}}\Vert \omega \Vert _{L^{2}}, \end{aligned}$$

and

$$\begin{aligned} \frac{1}{2}\frac{\hbox {d}}{\hbox {d}t}\Vert \omega \Vert ^{2}_{L^{2}}+4\chi \Vert \omega \Vert ^{2}_{L^{2}}= & {} 2\chi \int _{\mathbb {R}^{2}}\Omega \omega \hbox {d}x\hbox {d}y\\= & {} 2\chi \int _{\mathbb {R}^{2}}Z\omega \hbox {d}x\hbox {d}y+\frac{4\chi ^{4}}{\mu +\chi }\Vert \omega \Vert ^{2}_{L^{2}}\\\le & {} 2\chi \Vert Z\Vert _{L^{2}}\Vert \omega \Vert _{L^{2}}+\frac{4\chi ^{4}}{\mu +\chi } \Vert \omega \Vert ^{2}_{L^{2}}, \end{aligned}$$

therefore, we have

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t}\Vert Z\Vert _{L^{2}}+\frac{4\chi ^{2}}{\mu +\chi }\Vert Z\Vert _{L^{2}}\le \left( \frac{8\chi ^{2}}{\mu +\chi }-\frac{8\chi ^{3}}{(\mu +\chi )^{2}}\right) \Vert \omega \Vert _{L^{2}}, \end{aligned}$$

(4.2)

and

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t}\Vert \omega \Vert _{L^{2}}+\frac{4\mu \chi }{\mu +\chi }\Vert \omega \Vert _{L^{2}}\le 2\chi \Vert Z\Vert _{L^{2}}. \end{aligned}$$

(4.3)

Making $\mu =\chi$ and adding (4.2) and (4.3) together, we get

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t}\left( \Vert Z\Vert _{L^{2}}+\Vert \omega \Vert _{L^{2}}\right) \le 0. \end{aligned}$$

Integrating over the time interval [s, t], we obtain

$$\begin{aligned} \Vert Z(t)\Vert _{L^{2}}+\Vert \omega (t)\Vert _{L^{2}}\le \Vert Z(s)\Vert _{L^{2}}+\Vert \omega (s)\Vert _{L^{2}}. \end{aligned}$$

namely,

$$\begin{aligned} \Vert Z(t)\Vert ^{2}_{L^{2}}+\Vert \omega (t)\Vert ^{2}_{L^{2}}\le C\left( \Vert Z(s)\Vert ^{2}_{L^{2}}+\Vert \omega (s)\Vert ^{2}_{L^{2}}\right) . \end{aligned}$$

(4.4)

Using Minkowski’s inequality yields

$$\begin{aligned} \Vert Z(s)\Vert ^{2}_{L^{2}}= & {} \Vert \Omega -\frac{2\chi }{\mu +\chi }\omega \Vert ^{2}_{L^{2}}\\\le & {} C\left( \Vert \Omega (s)\Vert ^{2}_{L^{2}}+\Vert \omega (s)\Vert ^{2}_{L^{2}}\right) \\\le & {} C\left( \Vert \nabla u(s)\Vert ^{2}_{L^{2}}+\Vert \omega (s)\Vert ^{2}_{L^{2}}\right) . \end{aligned}$$

Multiplying the inequality (2.2) by $(1+t)^{n}$ and applying Theorem 1.1, we obtain

$$\begin{aligned}&\frac{\hbox {d}}{\hbox {d}t}\left[ (1+t)^{n}\left( \Vert u\Vert _{L^{2}}^{2}+\Vert \omega \Vert _{L^{2}}^{2}\right) \right] +2\varepsilon (1+t)^{n}\left( \Vert \nabla u\Vert ^{2}_{L^{2}}+\Vert \omega \Vert ^{2}_{L^{2}}\right) \\&\quad \le n(1+t)^{n-1}\left( \Vert u\Vert _{L^{2}}^{2}+\Vert \omega \Vert _{L^{2}}^{2}\right) \le C(1+t)^{n-2}, \end{aligned}$$

where $\varepsilon =\min \{\frac{\mu }{2},\,\frac{4\mu \chi }{\mu +2\chi }\}$. Then integrating the above inequality gets

$$\begin{aligned} \int _{0}^{t}(1+s)^{n}\left( \Vert \nabla u(s)\Vert ^{2}_{L^{2}}+\Vert \omega (s)\Vert ^{2}_{L^{2}}\right) \hbox {d}s \le C(1+t)^{n-1},\qquad \hbox {for}\quad \hbox {large}\ n>5,\nonumber \\ \end{aligned}$$

(4.5)

where C is the positive constant dependent on the initial data $u_{0},\, \omega _{0}$ and n.

Multiplying the inequality (4.4) by $(1+s)^{n}$ and integrating in $(\frac{t}{2}, t)$ with respect to s, we obtain

$$\begin{aligned}&\frac{t}{2}\left( 1+\frac{t}{2}\right) ^{n}\left( \Vert Z(t)\Vert ^{2}_{L^{2}} +\Vert \omega (t)\Vert ^{2}_{L^{2}}\right) \\&\quad \le \int _{\frac{t}{2}}^{t}(1+s)^{n}\left( \Vert Z(s)\Vert ^{2}_{L^{2}} +\Vert \omega (s)\Vert ^{2}_{L^{2}}\right) \hbox {d}s\\&\quad \le C\int _{\frac{t}{2}}^{t}(1+s)^{n}\left( \Vert \nabla u(s)\Vert ^{2}_{L^{2}}+\Vert \omega (s)\Vert ^{2}_{L^{2}}\right) \hbox {d}s\\&\quad \le C(1+t)^{n-1},\qquad \hbox {for}\quad \hbox {large} \ n>5, \end{aligned}$$

therefore, for $t\ge 1$, we have

$$\begin{aligned} \Vert Z(t)\Vert _{L^{2}}+\Vert \omega (t)\Vert _{L^{2}}\le C(1+t)^{-1}, \end{aligned}$$

which implies

$$\begin{aligned} \Vert \nabla u\Vert _{L^{2}}+\Vert \omega (t)\Vert _{L^{2}} \le C(1+t)^{-1}. \end{aligned}$$

Thus, we complete the proof of Theorem 1.3. $\square$