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A note on global existence of strong solution to the 3D micropolar equations with a damping term

Abstract

This paper studies the Cauchy problem of the 3D incompressible micropolar equations with a damping term \(\sigma |u|^{\beta -1}u\) (\(\sigma >0, 1\le \beta <3\)). It is shown that the strong solutions exist globally for any \(1\le \beta <3\).

Introduction

We consider the Cauchy problem of the 3D incompressible micropolar equations with a nonlinear damping term \(\sigma |u|^{\beta -1}u\) (\(\sigma >0, 1\le \beta <3\)) (see [5]):

$$ \textstyle\begin{cases} u_{t}+u\cdot \nabla u-(\nu +\kappa )\Delta u+\sigma \vert u \vert ^{ \beta -1}u+\nabla p=2\kappa \nabla \times w, \\ w_{t}+u\cdot \nabla w+4\kappa w-\gamma \Delta w-\mu \nabla {\operatorname{div}} w=2 \kappa \nabla \times u, \\ {\operatorname{div}}u=0, \\ u(x, 0)=u_{0}(x), \qquad w(x, 0)=w_{0}(x), \end{cases} $$
(1.1)

where \(u\in \mathbb{R}^{3}\), \(w\in \mathbb{R}^{3}\), \(p\in \mathbb{R}\) are the velocity field of fluid, the field of microrotation representing the angular velocity of the rotation of the fluid particles and the scalar pressure, respectively. The parameter ν is the kinematic viscosity; κ is the microrotation viscosity; γ and μ are the angular viscosities; σ is the damping coefficient.

When \(w=0\) and \(\kappa =0\), the system (1.1) is reduced to the incompressible damped Navier–Stokes equations which was studied firstly by Cai and Jiu [1]. They proved that the corresponding equations admit a global weak solution for any \(\beta \ge 1\) and a global strong solutions for \(\beta \ge \frac{7}{2}\). Moreover, the uniqueness was shown for any \(\frac{7}{2}\le \beta \le 5\). We refer to [3, 4, 68] for more results on the Navier–Stokes equations with a damping term.

Recently, the Cauchy problem (1.1) was considered by Ye [5]. It was proved that system (1.1) admits global strong solution for any \(\beta \ge 3\). In this paper, we aim to study existence of global solutions under some smallness condition of the initial data for any \(1\le \beta <3\). Before stating our main results, we firstly state the local strong solutions to (1.1), which can be proved by the similar technique as in [2]. Thus, we omit the details.

Theorem 1.1

Suppose that \(1\le \beta <3\), \((u_{0}, w_{0})\in H^{1}(\mathbb{R}^{3})\) with \(\operatorname{div}u_{0}=0\). Then there exist a small positive time \(T_{0}\) and a unique strong solution \((u, w)\) to the Cauchy problem (1.1) in \(\mathbb{R}^{3}\times (0, T_{0}]\).

Now, our main results read as follows.

Theorem 1.2

Assume that \((u_{0}, w_{0})\in H^{1}(\mathbb{R}^{3})\) with \(\operatorname{div}u_{0}=0\). \((u, w)(x, t)\) is the corresponding local strong to (1.1). For \(1\le \beta <3\), let \(T^{*}>0\) be a maximal existence time of the solution. If \(T^{*}<\infty \), then

$$\begin{aligned} \lim_{T\rightarrow T^{*}} \Vert u \Vert _{L^{s}(0, T; L^{r})}=\infty ,\quad \textit{with } \frac{2}{s}+\frac{3}{r}\le 1, 3< r< \infty . \end{aligned}$$
(1.2)

Theorem 1.3

Suppose \((u_{0}, w_{0})\in H^{1}(\mathbb{R}^{3})\) with \(\operatorname{div}u_{0}=0\). If \(1\le \beta <3\), then there exists a small positive constant \(\varepsilon _{0}\) depending only on μ, γ, σ, κ and ν, such that if

$$\begin{aligned} \bigl( \Vert w_{0} \Vert _{L^{2}}^{2}+ \Vert u_{0} \Vert _{L^{2}}^{2}\bigr) \bigl( \Vert \nabla w_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla u_{0} \Vert _{L^{2}}^{2}\bigr)\le \varepsilon _{0}, \end{aligned}$$
(1.3)

the Cauchy problem (1.1) admits a unique global strong solution, satisfying

$$\begin{aligned} &(u, w)\in L^{\infty }\bigl(0, T; H^{1}\bigl( \mathbb{R}^{3}\bigr)\bigr)\cap L^{2}\bigl(0, T; H^{2}\bigl( \mathbb{R}^{3}\bigr)\bigr), \\ & \vert u \vert ^{\frac{\beta -1}{2}}\nabla u\in L^{2}\bigl(0, T; L^{2}\bigl(\mathbb{R}^{3}\bigr)\bigr),\qquad \nabla \vert u \vert ^{\frac{\beta +1}{2}}\in L^{2}\bigl(0, T; L^{2}\bigl( \mathbb{R}^{3}\bigr)\bigr). \end{aligned}$$

Remark 1

When \(w=0\) and \(\kappa =0\), Theorem 1.1 and 1.2 generalize the previous results for the 3D Navier–Stokes equations with a damping term (see [7, 8]).

The proof of Theorem 1.2

This section is devoted to the proof of Theorem 1.2. In what follows, C denotes a generic positive constant depending only on μ, γ, σ, ν, κ and β. Let \((u, w)\) be a strong solution of (1.1) on \(\mathbb{R}^{3}\times (0, T)\) described in Theorem 1.1. As aforementioned, we shall prove Theorem 1.2 by contradiction arguments. So, from now on we assume otherwise that

$$\begin{aligned} \int _{0}^{T^{*}} \Vert u \Vert _{L^{r}}^{s}\,dt=M_{0}< \infty , \end{aligned}$$
(2.1)

with \(\frac{2}{s}+\frac{3}{r}\le 1\), \(3< r<\infty \).

First, multiplying (1.1)1 and (1.1)2 by u and w, respectively, integrating (by parts) the resulting equations over \(\mathbb{R}^{3}\), we have

$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \int \bigl( \vert u \vert ^{2}+ \vert w \vert ^{2}\bigr)\,dx+\sigma \int \vert u \vert ^{ \beta +1}\,dx+(\nu +\kappa ) \int \vert \nabla u \vert ^{2}\,dx \\ &\qquad {} +\gamma \int \vert \nabla w \vert ^{2}\,dx+4\kappa \int \vert w \vert ^{2}\,dx+\mu \vert { \operatorname{div}}w \vert ^{2}\,dx \\ &\quad =- \int u\cdot \nabla u\cdot u\,dx- \int u\cdot \nabla w\cdot w\,dx+2 \kappa \int (\nabla \times w)\cdot u\,dx+2\kappa \int (\nabla \times u) \cdot w\,dx \\ &\quad =2\kappa \int (\nabla \times w)\cdot u\,dx+2\kappa \int (\nabla \times u)\cdot w\,dx \\ &\quad =4\kappa \int (\nabla \times u)\cdot w\,dx\le \kappa \Vert \nabla u \Vert _{L^{2}}^{2}+4 \kappa \Vert w \Vert _{L^{2}}^{2}, \end{aligned}$$
(2.2)

which integrate with respect to t,

$$\begin{aligned} \bigl\Vert u(t) \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert w(t) \bigr\Vert _{L^{2}}^{2}+ \int _{0}^{t} \bigl(2\nu \Vert \nabla u \Vert _{L^{2}}^{2}+2\gamma \Vert \nabla w \Vert _{L^{2}}^{2} \bigr)\,dx\le \Vert u_{0} \Vert _{L^{2}}^{2}+ \Vert w_{0} \Vert _{L^{2}}^{2}. \end{aligned}$$
(2.3)

Next, multiplying (1.1)1 by \(-\Delta u\) and integrating (by parts) the resulting equations over \(\mathbb{R}^{3}\). By Hölder’s, Young’s and the Gagliardo–Nirenberg inequalities, we have

$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \int \vert \nabla u \vert ^{2}\,dx+\sigma \beta \int \vert u \vert ^{ \beta -1} \vert \nabla u \vert ^{2}\,dx +(\nu +\kappa ) \int \vert \Delta u \vert ^{2}\,dx \\ &\quad = \int u\cdot \nabla u\cdot \Delta u\,dx-2\kappa \int (\nabla \times w) \cdot \Delta u\,dx \\ &\quad \le \frac{\nu +\kappa }{4} \Vert \Delta u \Vert _{L^{2}}^{2}+ \frac{2}{\nu +\kappa } \int \vert u \vert ^{2} \vert \nabla u \vert ^{2}\,dx + \frac{8\kappa ^{2}}{\nu +\kappa } \Vert \nabla w \Vert _{L^{2}}^{2} \\ &\quad \le \frac{\nu +\kappa }{4} \Vert \Delta u \Vert _{L^{2}}^{2}+ \frac{C}{\nu +\kappa } \Vert u \Vert _{L^{r}}^{2} \Vert \nabla u \Vert _{L^{\frac{2r}{r-2}}}^{2} +\frac{8\kappa ^{2}}{\nu +\kappa } \Vert \nabla w \Vert _{L^{2}}^{2} \\ &\quad \le \frac{\nu +\kappa }{4} \Vert \Delta u \Vert _{L^{2}}^{2}+ \frac{8\kappa ^{2}}{\nu +\kappa } \Vert \nabla w \Vert _{L^{2}}^{2} + \frac{C}{\nu +\kappa } \Vert u \Vert _{L^{r}}^{2} \Vert \nabla u \Vert _{L^{2}}^{\frac{2(r-3)}{r}} \Vert \Delta u \Vert _{L^{2}}^{\frac{6}{r}} \\ &\quad \le \frac{\nu +\kappa }{2} \Vert \Delta u \Vert _{L^{2}}^{2}+ \frac{8\kappa ^{2}}{\nu +\kappa } \Vert \nabla w \Vert _{L^{2}}^{2} + \frac{C}{(\nu +\kappa )^{2}} \Vert u \Vert _{L^{r}}^{\frac{2r}{r-3}} \Vert \nabla u \Vert _{L^{2}}^{2} \end{aligned}$$
(2.4)

with arbitrary \(r>3\).

Similarly, multiplying (1.1)2 by \(-\Delta w\) and integrating (by parts) the resulting equations over \(\mathbb{R}^{3}\), we have

$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \int \vert \nabla w \vert ^{2}\,dx+\gamma \int \vert \Delta w \vert ^{2}\,dx+4 \kappa \int \vert \nabla w \vert ^{2}\,dx+\mu \vert \nabla { \operatorname{div}}w \vert ^{2}\,dx \\ &\quad = \int u\cdot \nabla w\cdot \Delta w\,dx-2\kappa \int (\nabla \times u) \cdot \Delta w\,dx \\ &\quad \le \frac{\gamma }{4} \Vert \Delta w \Vert _{L^{2}}^{2}+ \frac{2}{\gamma } \int \vert u \vert ^{2} \vert \nabla w \vert ^{2}\,dx+\frac{8\kappa ^{2}}{\gamma } \Vert \nabla u \Vert _{L^{2}}^{2} \\ &\quad \le \frac{\gamma }{2} \Vert \Delta w \Vert _{L^{2}}^{2}+ \frac{8\kappa ^{2}}{\gamma } \Vert \nabla u \Vert _{L^{2}}^{2}+ \frac{C}{\gamma ^{2}} \Vert u \Vert _{L^{r}}^{\frac{2r}{r-3}} \Vert \nabla w \Vert _{L^{2}}^{2}. \end{aligned}$$
(2.5)

Adding (2.4) and (2.5), by Gronwall’s inequality and (2.3), we have

$$\begin{aligned} \Vert \nabla u \Vert _{L^{2}}^{2}+ \Vert \nabla w \Vert _{L^{2}}^{2}+ \int _{0}^{t}\bigl( \Vert \Delta u \Vert _{L^{2}}^{2}+ \Vert \Delta w \Vert _{L^{2}}^{2} \bigr)\le C\bigl( \Vert \nabla u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla w_{0} \Vert _{L^{2}}^{2}\bigr). \end{aligned}$$
(2.6)

Therefore, if \(u\in L^{s}(0, T; L^{r})\) with \(\frac{2}{s}+\frac{3}{r}\le 1\), we can take \((u, w)|_{t=T^{*}}\) as the initial data, then the local strong solutions \((u, w)\) can be extended beyond \(T^{*}\). This contradicts the assumption that \(T^{*}>0\) is the maximal existence time. The proof of Theorem 1.2 is complete.

The proof of Theorem 1.3

Throughout this section, we denote

$$\begin{aligned} C_{0}:= \Vert u_{0} \Vert _{L^{2}}^{2}+ \Vert w_{0} \Vert _{L^{2}}^{2}. \end{aligned}$$
(3.1)

Let \((u, w)\) be the strong solution to the problem (1.1) on \(\mathbb{R}^{3}\times (0, T)\), then one has the following estimates. First, we infer from (2.4) and (2.5) that

$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \int \bigl( \vert \nabla u \vert ^{2}+ \vert \nabla w \vert ^{2} \bigr)\,dx+ \sigma \beta \int \vert u \vert ^{\beta -1} \vert \nabla u \vert ^{2}\,dx +(\nu +\kappa ) \int \vert \Delta u \vert ^{2}\,dx \\ &\qquad {} +\gamma \int \vert \Delta w \vert ^{2}\,dx+4\kappa \int \vert \nabla w \vert ^{2}\,dx+ \mu \vert \nabla { \operatorname{div}}w \vert ^{2}\,dx \\ &\quad = \int u\cdot \nabla u\cdot \Delta u\,dx+ \int u\cdot \nabla w\cdot \Delta w\,dx-4\kappa \int (\nabla \times w)\cdot \Delta u\,dx \\ &\quad \le 4\kappa \Vert \nabla w \Vert _{L^{2}}^{2}+ \biggl(\kappa +\frac{\nu }{4}\biggr) \Vert \Delta u \Vert _{L^{2}}^{2}+\frac{\gamma }{2} \Vert \Delta w \Vert _{L^{2}}^{2} \\ &\qquad {}+ \frac{1}{\nu } \int \vert u \vert ^{2} \vert \nabla u \vert ^{2}\,dx +\frac{1}{2\gamma } \int \vert u \vert ^{2} \vert \nabla w \vert ^{2}\,dx \\ &\quad \le 4\kappa \Vert \nabla w \Vert _{L^{2}}^{2}+ \biggl(\kappa +\frac{\nu }{4}\biggr) \Vert \Delta u \Vert _{L^{2}}^{2}+\frac{\gamma }{2} \Vert \Delta w \Vert _{L^{2}}^{2}+ \frac{1}{\nu } \Vert u \Vert _{L^{\infty }}^{2} \Vert \nabla u \Vert _{L^{2}}^{2} \\ &\qquad {} + \frac{1}{2\gamma } \Vert u \Vert _{L^{\infty }}^{2} \Vert \nabla w \Vert _{L^{2}}^{2} \\ &\quad \le 4\kappa \Vert \nabla w \Vert _{L^{2}}^{2}+ \biggl(\kappa +\frac{\nu }{4}\biggr) \Vert \Delta u \Vert _{L^{2}}^{2}+\frac{\gamma }{2} \Vert \Delta w \Vert _{L^{2}}^{2} \\ &\qquad {} +C \Vert u \Vert _{L^{6}} \Vert \Delta u \Vert _{L^{2}} \biggl(\frac{1}{\nu } \Vert \nabla u \Vert _{L^{2}}^{2}+ \frac{1}{2\gamma } \Vert \nabla w \Vert _{L^{2}}^{2} \biggr) \\ &\quad \le 4\kappa \Vert \nabla w \Vert _{L^{2}}^{2}+ \biggl(\kappa +\frac{\nu }{2}\biggr) \Vert \Delta u \Vert _{L^{2}}^{2}+\frac{\gamma }{2} \Vert \Delta w \Vert _{L^{2}}^{2}+ \frac{C}{\nu ^{3}} \Vert \nabla u \Vert _{L^{2}}^{6} \\ &\qquad {} +\frac{C}{\gamma ^{3}} \Vert \nabla u \Vert _{L^{2}}^{2} \Vert \nabla w \Vert _{L^{2}}^{4}. \end{aligned}$$
(3.2)

Then we obtain after integrating (3.2) with respect to t

$$\begin{aligned} &\sup_{0\le s\le t} \bigl( \Vert \nabla u \Vert _{L^{2}}^{2}+ \Vert \nabla w \Vert _{L^{2}}^{2} \bigr)+ \int _{0}^{t} \bigl( \Vert \Delta u \Vert _{L^{2}}^{2}+ \Vert \Delta w \Vert _{L^{2}}^{2} \bigr)\,ds \\ &\qquad {} + \int _{0}^{t} \biggl(\sigma \bigl\Vert \vert u \vert ^{\frac{\beta -1}{2}}\nabla u \bigr\Vert _{L^{2}}^{2} + \frac{4\sigma (\beta -1)}{(\beta +1)^{2}} \bigl\Vert \nabla \vert u \vert ^{\frac{\beta +1}{2}} \bigr\Vert _{L^{2}}^{2} \biggr)\,ds \\ &\quad \le \Vert \nabla u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla w_{0} \Vert _{L^{2}}^{2}+C_{1} \int _{0}^{t} \bigl( \Vert \nabla u \Vert _{L^{2}}^{6}+ \Vert \nabla u \Vert _{L^{2}}^{2} \Vert \nabla w \Vert _{L^{2}}^{4}\bigr)\,ds \\ &\quad \le \Vert \nabla u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla w_{0} \Vert _{L^{2}}^{2}+C_{1} \sup_{0\le s\le t} \bigl( \Vert \nabla u \Vert _{L^{2}}^{4}+ \Vert \nabla w \Vert _{L^{2}}^{4}\bigr) \int _{0}^{t} \Vert \nabla u \Vert _{L^{2}}^{2}\,ds \\ &\quad \le \Vert \nabla u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla w_{0} \Vert _{L^{2}}^{2}+C_{1}C_{0} \sup_{0\le s\le t} \bigl( \Vert \nabla u \Vert _{L^{2}}^{4}+ \Vert \nabla w \Vert _{L^{2}}^{4}\bigr). \end{aligned}$$
(3.3)

Next, define the function \(A(t)\) as follows:

$$\begin{aligned} A(t):=\sup_{0\le s\le t} \bigl( \Vert \nabla u \Vert _{L^{2}}^{2}+ \Vert \nabla w \Vert _{L^{2}}^{2} \bigr). \end{aligned}$$
(3.4)

Due to the regularity of u and w, one can deduce that \(A(t)\) is a continuous function on \([0, T]\). According to (3.3), we have

$$\begin{aligned} A(t)\le \Vert \nabla u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla w_{0} \Vert _{L^{2}}^{2}+C_{1}C_{0}A^{2}(t). \end{aligned}$$
(3.5)

Now, by (3.5), one can prove that

$$\begin{aligned} A(t)\le 3 \bigl( \Vert \nabla u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla w_{0} \Vert _{L^{2}}^{2} \bigr). \end{aligned}$$
(3.6)

In fact, we assume that

$$\begin{aligned} \bigl( \Vert u_{0} \Vert _{L^{2}}^{2}+ \Vert w_{0} \Vert _{L^{2}}^{2}\bigr) \bigl( \Vert \nabla u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla w_{0} \Vert _{L^{2}}^{2}\bigr)\le \varepsilon _{0} \end{aligned}$$
(3.7)

and set

$$\begin{aligned} T_{*}:=\max \bigl\{ t\in [0, T]: A(t)\le 3 \bigl( \Vert \nabla u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla w_{0} \Vert _{L^{2}}^{2} \bigr), \forall s\in [0, t] \bigr\} . \end{aligned}$$
(3.8)

Then we claim that

$$\begin{aligned} T=T_{*}. \end{aligned}$$

Otherwise, we have \(T_{*}\in (0, T)\). By the continuity of \(A(t)\), it follows from (3.5), (3.7)–(3.8) that

$$\begin{aligned} A(T_{*})&\le \Vert \nabla u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla w_{0} \Vert _{L^{2}}^{2}+C_{1}C_{0}A^{2}(T_{*}) \\ &\le \Vert \nabla u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla w_{0} \Vert _{L^{2}}^{2}+3C_{1}C_{0} \bigl( \Vert \nabla u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla w_{0} \Vert _{L^{2}}^{2} \bigr)A(T_{*}) \\ &\le \Vert \nabla u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla w_{0} \Vert _{L^{2}}^{2}+ \frac{1}{2}A(T_{*}), \end{aligned}$$
(3.9)

here, we choose \(\varepsilon _{0}=\frac{1}{6C_{1}}\). Thus, from (3.9) we deduce that

$$\begin{aligned} A(T_{*})\le 2 \bigl( \Vert \nabla u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla w_{0} \Vert _{L^{2}}^{2} \bigr). \end{aligned}$$
(3.10)

This contradicts (3.8). Hence, by virtue of the argument of continuity and (3.10), we can easily get the desired (3.6).

Finally, we ready to give the proof of Theorem 1.2. In fact, due to Theorem 1.1, there is a unique local strong solution \((u, w)\) to Eqs. (1.1). Let \(T^{*}\) be the maximal existence time to the solution. We will show that \(T^{*}=\infty \). Otherwise, by contradiction, we take \(T^{*}<\infty \), then, by Theorem 1.2, we get, for any \((s, r)\) with \(\frac{2}{s}+\frac{3}{r}\le 1\), \(3< r<\infty \),

$$\begin{aligned} \int _{0}^{T^{*}} \Vert u \Vert _{L^{r}}^{s}\,dt=\infty , \end{aligned}$$
(3.11)

which together with Sobolev’s inequality \(\|u\|_{L^{6}}\le C\|\nabla u\|_{L^{2}}\) leads to

$$\begin{aligned} \int _{0}^{T^{*}} \Vert \nabla u \Vert _{L^{2}}^{4}\,dt=\infty . \end{aligned}$$
(3.12)

On the other hand, by Hölder’s inequality, the Gagliardo–Nirenberg inequality, (2.3) and (3.7), we get

$$\begin{aligned} \int _{0}^{T^{*}} \Vert \nabla u \Vert _{L^{2}}^{4}\,dt&\le \sup_{0\le t\le T^{*}} \Vert \nabla u \Vert _{L^{2}}^{2} \int _{0}^{T^{*}} \Vert \nabla u \Vert _{L^{2}}^{2}\,dt \\ &\le C \bigl( \Vert u_{0} \Vert _{L^{2}}^{2}+ \Vert w_{0} \Vert _{L^{2}}^{2} \bigr) \bigl( \Vert \nabla u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla w_{0} \Vert _{L^{2}}^{2} \bigr) \\ &< +\infty , \end{aligned}$$
(3.13)

contradicting (3.12). This contradiction shows that \(T^{*}=\infty \), and thus we obtain the global strong solution of (1.1). This ends the proof of Theorem 1.3.

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The authors would like to thank the anonymous reviewers and the Editor for their constructive comments which helped to improve the quality of the paper.

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WW and YL designed the methodology. WW wrote the draft and derived the theorem. YL reviewed and revised the paper. All authors read and approved the final manuscript.

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Wang, W., Long, Y. A note on global existence of strong solution to the 3D micropolar equations with a damping term. Bound Value Probl 2021, 72 (2021). https://doi.org/10.1186/s13661-021-01548-z

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Keywords

  • Micropolar equations
  • Damping
  • Global regularity