Logarithmically extended global regularity result of Lans-alpha MHD system in two-dimensional space

We study the two-dimensional generalized Lans alpha magnetohydrodynamics system. We show that the solution pairs of velocity and magnetic fields to this system preserve their initial regularity in two cases: dissipation logarithmically weaker than a full Laplacian and zero diffusion, zero dissipation and diffusion logarithmically weaker than a full Laplacian.