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arXiv · 1203.0990

On a singular incompressible porous media equation

Susan Friedlander·Francisco Gancedo·Weiran Sun·Vlad Vicol
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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.

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Susan Friedlander, Francisco Gancedo, Weiran Sun, Vlad Vicol. 2012-03-05. On a singular incompressible porous media equation. https://doi.org/10.1063/1.4725532

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