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 (−∂22βb1,−∂11βb2) with β>1. In the second case, the velocity has a fractional Laplacian dissipation (−Δ)αu with any α>0, the micro-rotational velocity is with Laplacian dissipation and the magnetic field has partial diffusion (−∂22b1,−∂11b2). In two cases the global well-posedness of classical solutions is proved in this paper.