  In this paper we study a singularly modified version of the incompressible
porous media equation. We investigate the implications for the local
well-posedness of the equations by modifying, with a fractional derivative, the
constitutive relation between the scalar density and the convecting divergence
free velocity vector. Our analysis is motivated by recent work \cite{CCCGW}
where it is shown that for the surface quasi-geostrophic equation such a
singular modification of the constitutive law for the velocity, quite
surprisingly still yields a locally well-posed problem. In contrast, for the
singular active scalar equation discussed in this paper, local well-posedness
does not hold for smooth solutions, but it does hold for certain weak
solutions.
