Global regularity for 2D magneto-micropolar equations with only micro-rotational velocity dissipation and magnetic diffusion

Abstract

This paper studies the global regularity of classical solutions to 2D magneto-micropolar fluid equations with only micro-rotational velocity dissipation and magnetic diffusion. Here the micro-rotational velocity dissipation and magnetic diffusion are given by −ΔΩ and (−Δ)βb. Making use of several combined quantities, maximal regularity of heat operator and Littlewood–Paley decomposition theory, we establish a regularity criterion in terms of magnetic field for the case β=1 and the global regularity for β>1. The regularity criterion given here is also new even for the 2D magnetohydrodynamic equations. In addition, to prove these two main results, as preparation we establish a new global a priori estimate for magnetic field, namely Δb∈L∞(0,T;Lp(R2)) with p≥2 which also holds for the 2D magnetohydrodynamic equations as a particular case.