Global regularity of NN-dimensional generalized MHD system with anisotropic dissipation and diffusion

Motivated by the anisotropic Navier–Stokes equations that have much applications in studying fluids in thin domains, in particular meteorology and oceanography, we study the magnetohydrodynamics system with generalized dissipation and diffusion, considering different exponents of the fractional Laplacians with logarithmic worsening applied to different directions and components of the solution vector fields. The results indicate that it is possible to regularize the flows by anisotropic dissipation, some components in various directions allowed to be below the critical exponents in the expense of others being above.