Smooth small perturbations of the explicit IPM self-similar blow-up still blow up in the same self-similar form, with a sharp regularity threshold at C^2.
Finite time blow-up in a 1D model of the incompressible porous media equation
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We derive a PDE that models the behavior of a boundary layer solution to the incompressible porous media (IPM) equation posed on the 2D periodic half-plane. This 1D IPM model is a transport equation with a non-local velocity similar to the well-known C\'{o}rdoba-C\'{o}rdoba-Fontelos (CCF) equation. We discuss how this modification of the CCF equation can be regarded as a reasonable model for solutions to the IPM equation. Working in the class of bounded smooth periodic data, we then show local well-posedness for the 1D IPM model as well as finite time blow-up for a class of initial data.
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Stable self-similar singularity formation for infinite energy solutions of the incompressible porous medium equations
Smooth small perturbations of the explicit IPM self-similar blow-up still blow up in the same self-similar form, with a sharp regularity threshold at C^2.