Finite time blow-up in a 1D model of the incompressible porous media equation

TL;DR

This paper introduces a 1D PDE model for the boundary layer of the incompressible porous media equation, demonstrating finite time blow-up for certain initial conditions, thus providing insights into the model's singularity formation.

Contribution

It derives a new 1D PDE model resembling the CCF equation for IPM boundary layers and proves finite time blow-up for specific initial data.

Findings

Finite time blow-up occurs for certain initial conditions.

The 1D model is well-posed locally in time.

The model captures key features of the boundary layer behavior.

Abstract

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.