New regularity criteria for an MHD Darcy-Forchheimer fluid

The purpose of the presented article is to develop some new global regularity criteria for a magnetohydrodynamic (MHD) fluid flowing in a saturated porous medium. The effect of the porous medium, over the fluid flow, is characterized by a Darcy–Forchheimer law. The fluid, under study, is considered as one-dimensional and flowing in the x–direction with velocity component u. In addition, such a component is assumed to vary with the y–direction, i.e. u(y). Then, given the vorticity function $w=-\frac{\partial u}{\partial y}$, such that ${\Vert w\Vert }_{\text{BMO}}^2$ is sufficiently small, we develop the regularity criteria under the scope of the L² space. We extend our results to the spaces L^s, where s > 2. Afterward, we prove the Liouville-type theorem for the MHD Darcy–Forchheimer flow equation. Eventually, we obtain some characterization about the asymptotic behaviour of solutions, particularly, the nonuniform convergence in L² for $t\to \infty$.