Global regularity criteria for the 2D Magneto-micropolar Equations with Partial Dissipation

Abstract

The magneto-micropolar equations model the motion of electrically conducting micropolar fluids in the presence of a magnetic field.These equations have been the focus of numerous analytical, experimental, and numerical investigations.One fundamental problem concerning these equations is whether their classical solutions are globally regular for all time or if they develop finite time singularities.The global regularity problem can be particularly challenging when there is only partial dissipation. In this paper, we study the 2D incompressible magneto-micropolar equations with partial dissipation prove two new regularity results. The first result addresses a weak solution, and the second result establishes global regularity criteria. As a consequence, we can single out one special partial dissipation case and establish the global regularity if (∂y u1, ∂y u2) ∈ L∞ [0, T ], R2. The proofs of our main results rely on anisotropic Sobolev-type inequalities and the appropriate combination and cancellation of terms.