We prove several Liouville type results for the stationary MHD and Hall-MHD equations. In particular, we show that the velocity and magnetic field, belonging to some Lorentz spaces or satisfying a priori decay assumption, must be zero.

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In this paper, we consider the regularity of weak solutions to the 3D magneto‐micropolar fluid equations. It is shown that if the velocity field or pressure belongs to some Lorentz spaces in both time and spatial directions, then the weak solutions are regular on [0, T]. In particular, we also obtain regularity criteria for the micropolar fluid equations and the MHD equations, respectively. Our results extend and generalize previous results.

This paper is concerned with the Liouville type theorems for the 3D stationary incompressible Hall-magnetohydrodynamic (Hall-MHD) equations. We establish that under some sufficient conditions in local Morrey spaces, solutions of the stationary Hall-MHD equations are identically zero. In particular, we also prove Liouville type results for the stationary incompressible MHD equations on R 3 . Our theorems extend and generalize the classical results for the stationary incompressible Navier-Stokes equations.

Studied in this paper is the Cauchy problem for the 3D incompressible Hall-MHD system with horizontal dissipation. It is shown that if the initial data is axisymmetric and the swirl component of the velocity and the magnetic vorticity are trivial, such a system is globally well-posed for the large initial data. The key is to take full advantage of the structure of the Hall-MHD system in axisymmetric case to overcome the main difficulty due to the absence of vertical dissipation.