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.

Keywords

porous media equation, Darcy’s law, regularity criteria

2010 Mathematics Subject Classification

35B65, 35Q35, 35Q86

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Published 3 December 2014