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The 3D generalized incompressible magneto-micropolar equations with fractional dissipation can be written as: [
① https://doi.org/10.1016/j.camwa.2018.08.029,
② Shang, H., Wu, J. Global regularity for 2D fractional magneto-micropolar equations. Math. Z. 297, 775–802 (2021).
③ Magneto-micropolar fluid motion: global existence of strong solutions (in bounded domain).
④ The magneto-micropolar equations with periodic boundary conditions: Solution properties at potential blow-up times (in torus)]

\begin{align} {\label{1}
(1) \left\{ \begin{array}{ll} \partial _t u + u \cdot \nabla u +(\nu +\kappa ) (-\Delta )^{\alpha } u = -\nabla p +2 \kappa \nabla \times \Omega + b\cdot \nabla b,\\
\partial _t \Omega + u\cdot \nabla \Omega-\lambda \nabla \div \Omega + 4\kappa \Omega + \mu \, (-\Delta )^{\gamma } \Omega = 2 \kappa \nabla \times u, \\
\partial _t b + u\cdot \nabla b + \eta (-\Delta )^{\beta } b= b\cdot \nabla u, \\
\nabla \cdot u =0, ~ \nabla \cdot b =0,\\
(u(0,x), \Omega (0,x), b(0,x)) =(u_0(x), \Omega _0(x), b_0(x)). \end{array}\right.
}
\end{align}

for $t>0$ and $x\in \R^3$. We denote $u(t,x)$, $\Omega(t,x)$, $b(t,x)$, $p(t,x)$ the velocity field, the micro-rotational velocity, the magnetic field and the hydrostatic pressure, respectively. $\nu$ denotes the kinematic viscosity, $\kappa$ the vortex viscosity, $1/\eta$ the magnetic Reynolds number, $\mu$ and $\lambda$ the angular viscosities. $\alpha$, $\beta$, and $\gamma$ are the given constants. The fractional Laplacian operator $(-\Delta )^\alpha$ is defined via the Fourier transform

$$\begin{aligned} \widehat{(-\Delta )^\alpha f} (\xi ) = |\xi |^{2\alpha } \, \widehat{f}(\xi ). \end{aligned}$$ When $\alpha =\beta =\gamma =1$, (1) becomes the magneto-micropolar equations with standard Laplacian operator dissipation. When the magnetic field $b=0$, the system (1) reduces to the micropolar fluid equations, which describe some physical phenomena such as the motion of animal blood, liquid crystals and dilute aqueous polymer solutions. We define the 2D magneto-micropolar equations by setting

$$u=(u_1,u_2,0),\quad b=(b_1,b_2,0),\quad \Omega=(0,0,\Omega).$$

More explicitly, the 2D magneto-micropolar system is expressed as

\begin{align} {\label{2}
\left\{ \begin{array}{ll} \partial _t u + u \cdot \nabla u +(\nu +\kappa ) (-\Delta )^{\alpha } u = -\nabla p +2 \kappa \nabla \times \Omega + b\cdot \nabla b,\\
\partial _t \Omega + u\cdot \nabla \Omega+ 4\kappa \Omega + \mu \, (-\Delta )^{\gamma } \Omega = 2 \kappa \nabla \times u, \\
\partial _t b + u\cdot \nabla b + \eta (-\Delta )^{\beta } b= b\cdot \nabla u, \\
\nabla \cdot u =0, ~ \nabla \cdot b =0,\\
(u(0,x), \Omega (0,x), b(0,x)) =(u_0(x), \Omega _0(x), b_0(x)). \end{array}\right.
}
\end{align}

Micropolar fluids represent a class of fluids with nonsymmetric stress tensor (called polar fluids) such as fluids consisting of suspending particles, dumbbell molecules, etc. The magneto-micropolar equations model the motion of electrically conducting micropolar fluids in the presence of a magnetic field. They govern a wide range of fluids such as the motion of aggregates of small solid ferromagnetic particles in viscous magnetic fluids. The magneto-micropolar equations are derived by combining the equations of continuity, momentum, Maxwell and angular momentum (see, e.g., [Berkovski, B., Bashtovoy, V.: Magnetic Fluids and Applications Handbook. (1996), https://doi.org/10.1002/mma.1170]).