This paper deals with the global existence and decay estimates of
solutions to the three-dimensional magneto-micropolar fluid equations
with only velocity dissipation and magnetic diffusion in the whole space
with various Sobolev and Besov spaces. Specifically, we first
investigate the global existence and optimal decay estimates of weak
solutions. Then we prove the global existence of solutions with small
initial data in $H^s$, $B_{2, \infty}^s$ and
critical Besov spaces, respectively. Furthermore, the optimal decay
rates of these global solutions are correspondingly established in
$\dot{H}^m$ and $\dot{B}_{2,
\infty}^m$ spaces with $0\leq
m\leq s$ and in $\dot{B}_{2,
1}^{m}$ with $0\leq m\leq
\frac 12$, when the initial data belongs to
$\dot{B}_{2, \infty}^{-l}$
($0< l\leq\frac32$). The main
difficulties lie in the presence of linear terms and the lack of
micro-rotation velocity dissipation. To overcome them, we make full use
of the special structure of the system and employ various techniques
involved with the energy methods, the improved Fourier splitting,
Fourier analysis and the regularity interpolation methods.
Global well-posedness and optimal time decay rates of solutions to the three-dimensional magneto-micropolar fluid equations