Global regularity for the 2D magneto-micropolar system with partial and fractional dissipation

This paper focuses on the 2D incompressible magneto-micropolar sysytem with the kinematic dissipation given by the fractional operator (−Δ)^α, the magnetic diffusion by the fractional operator (−Δ)^β and the spin dissipation by the fractional operator (−Δ)^γ. α,β, and γ are nonnegative constants. We proved that this system with any α+β=2,1 ≤ α ≤ 2,γ=0, and α+γ ≥ 1,β=1 always possesses a unique global smooth solution $(\mathbf{u},\mathbf{b},\mathrm{w})\in H^s({\mathbb{R}}^2)(s>2)$ if the initial data is sufficiently smooth. In addition, we also obtained the global regularity results for several partial dissipation cases.