Large-time behavior of solutions to the 2D generalized magneto-micropolar equations

This work provides rigorous verification of the super-smoothing effect of higher-order fractional Laplacian dissipation. Shang and Zhao (2017) have proved the global regularity of classical solutions of the 2D incompressible magneto-micropolar equations with linear velocity damping $u$, microrotational dissipation $(-\mathrm{\Delta })\omega$, and magnetic diffusion $(-\mathrm{\Delta }{)}^{\beta }b,\beta >1$. This paper is devoted to further investigating the large-time behavior of global smooth solutions of the system with $1<\beta \le \frac{3}{2}$. We apply the negative Sobolev space to overcome the difficulty caused by fractional-order dissipation and establish ${\int }_0^t\Vert \mathrm{\nabla }b(\tau ){\Vert }_{L^2}d\tau \le C$. Furthermore, by fully exploiting the special structure of the system and combining the properties of a heat operator with the generalized Fourier splitting methods, we obtain the decay rates of the solutions and their first-order derivatives for $1\le p\le \frac{2}{\beta }$.