Global Regularity Of The 2D Incompressible Generalized Magneto-micropolar Equations

This paper studies 2D incompressible generalized magnetic-micropolar fluid equa-tions:where u=(u1,u2,0),b=(b1,b2,0),=(0,0,?).Magnetic micropolar fluids describe the motion of conductive micropolar fluids under magnetic field.In recent years,the global regularity of the 2D magneto-micropolar equations has attracted extensive attention in mathematics and many achievements have been achieved.However,when?=?=1/2,?=v=0,?=1 or ?=-1/2,?=1/2,??v=1,??v=1,the global regularity of the 2D incompressible magneto-micropolar fluid equations(0.0.2)is still an open problem.In this paper,we are committed to improving the previous results and establishing the global regularity of the Leray-? regularized models for the above open problems.Chapter three studies two cases of global regularity problems on the 2D magneto-micropolar equations with fractional partial diffusion.For the first case the velocity is ideal fluid,the micro-rotational velocity is with Laplacian dissipation and the mag-netic field has fractional partial diffusion (?) with ?>1.In the second case,the velocity has a fractional Laplacian dissipation(-?)?u with any ?>0,the micro-rotational velocity is with Laplacian dissipation and the magnetic field has partial diffusion (?) In two cases the global well-posedness of classical solution are proved in this dissertation.In Chapter four,we consider the Cauchy problem of the 2D Leray-? regularized in-compressible magneto-micropolar equations.The global smooth solutions of the Cauchy problem for the equations with only velocity dissipation or with only angular viscosity and magnetic diffusion are obtained.