This paper is devoted to the study of the Liouville-type theorem of the stationary fractional compressible MHD systems in anisotropic Lebesgue spaces in ${\mathbb{R}}^3$. We showed that the solution is trivial when certain anisotropic integrability conditions are satisfied in terms of the velocity and the magnetic field components.

This paper concerns the regularity criteria of the three-dimensional magnetohydrodynamic (MHD) system in anisotropic Lebesgue spaces. Two regularity results were proved under additional assumptions on the horizontal components of the velocity field $\mathbf{u}$ and the magnetic field $\mathbf{B}$, or directions of Elsässer's variables $\mathbf{u}\pm \mathbf{B}$.