Regularity criteria of the porous media equation in terms of one partial derivative or pressure field

We obtain new regularity criteria and smallness conditions for the global regularity of the N-dimensional supercritical porous media equation. In particular, it is shown that in order to obtain global regularity result, one only needs to bound a partial derivative in one direction or the pressure scalar field. Our smallness condition is also in terms of one direction, dropping conditions on (N -1) other directions completely, or the pressure scalar field. The proof relies on key observations concerning the incompressibility of the velocity vector field and the special identity derived from Darcy's law.