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

Abstract

This paper studies two cases of global regularity problems on the 2D magneto-micropolar equations with partial magnetic diffusion and fractional dissipation. For the first case the velocity field is ideal, the micro-rotational velocity is with Laplacian dissipation and the magnetic field has fractional partial diffusion $(-{\partial }_{22}^{\beta }{b}_1,-{\partial }_{11}^{\beta }{b}_2)$ with $\beta >1$. In the second case, the velocity has a fractional Laplacian dissipation ${(-\Delta )}^{\alpha }u$ with any $\alpha >0$, the micro-rotational velocity is with Laplacian dissipation and the magnetic field has partial diffusion $(-{\partial }_{22}{b}_1,-{\partial }_{11}{b}_2)$. In two cases the global well-posedness of classical solutions is proved in this paper.