The Regularity Criteria For The Magneto-micropolar Fluid Equations

Since Navier-Stokes been raised, the hydrodynamic equations have been one of the hot issues in the research of partial differential equations. This work has very important theoretical significance and practical value in mechanics, control the process, ecological and economic system, astronomy, and so on. In recent years, micropolar fluid equations attracted the attention of more and more scholars.In this paper, we consider the Cauchy problems of the2D and3D incompressible magneto-micropolar fluid equations, which describe the motion of a micropolar fluid in the presence of a magnetic field.In the case of3D, the global existence of smooth solution is still a challenging open problem. Due to the lack of global well-posedness theory, the establishment of global regularity is very important in both theory and practice. Therefore, many scholars turned to make a series of studies for its global regularity, and got a number of important conclusions.Based on these conclusions, by using the methods of Cao-Wu and standard energy estimates, we established two sufficient conditions for extending the solution of3D magneto-micropolar fluid equations:(1) if (u0,w0,b0)∈H3(R3)with divu0=divb0=0, for some T>0,(u,w,b) is the corresponding local smooth solution to the magneto-micropolar fluid equations on [0, T), and the fluid velocity u satisfies(2) if (u0,w0,b0)∈H1(R3)∩L4(R3)with divu0=divb0=0, for some T>0(u,w,b)is the corresponding local smooth solution to the magneto-micropolar fluid equations on[0, T), the pressure π and the magnetic field b satisfiesIn the case of2D, inspired by the work of Dong-Zhang, by using Littlewood-Paley decomposition and classic Friedrichs method-approximate the equations by a cut-off in the frequency space, we proposed the local existence and uniqueness of the smoothsolution of2D magneto-micropolar fluids with zero angular viscosity.