| Abstract: | This paper studies the global existence and regularity of classical solutions
to the 2D incompressible magneto-micropolar equations with partial dissipation. The
magneto-micropolar equations model the motion of electrically conducting micropolar
fluids in the presence of a magnetic field. When there is only partial dissipation, the
global regularity problem can be quite difficult. We are able to single out three special
partial dissipation cases and establish the global regularity for each case. As special
consequences, the 2D Navier-Stokes equations, the 2D magnetohydrodynamic equations,
and the 2D micropolar equations with several types of partial dissipation always
possess global classical solutions. The proofs of our main results rely on anisotropic
Sobolev type inequalities and suitable combination and cancellation of terms. |
