On a singular incompressible porous media equation

TL;DR

This paper explores a singular modification of the incompressible porous media equation using fractional derivatives, revealing that local well-posedness holds for some weak solutions but not for smooth ones.

Contribution

It introduces a fractional derivative modification to the porous media equation and analyzes its impact on local well-posedness, contrasting with similar modifications in related equations.

Findings

Local well-posedness fails for smooth solutions.

Weak solutions exhibit local well-posedness.

Modification impacts the equation's mathematical properties.

Abstract

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.