Recent Developments on the Micropolar and Magneto-Micropolar Fluid Systems: Deterministic and Stochastic Perspectives

Abstract

We review recent developments on the micropolar and magneto-micropolar fluid systems in spatial dimensions two and three from both deterministic and stochastic perspectives. Under the deterministic setting, we review the global regularity result in two-dimensional space with zero angular viscosity and a regularity criterion in three-dimensional space that involves only two velocity vector field components. Under the stochastic setting, we review the existence of a weak martingale solution in three-dimensional space and the unique strong solution in two-dimensional space under a suitable condition on the noise. Throughout the paper, we compare these results with other partial differential equations related to fluid mechanics, such as the Navier-Stokes equations, magnetohydrodynamics and Boussinesq systems.