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

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 Córdoba-Córdoba-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.

14 pages, 1 figure; V2 includes minor edits

References in corpus (5)

• Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data I: Analysis
• The 2D Muskat Problem II: Stable Regime Small Data Singularity on the Half-plane
• The 2D Muskat Problem I: Local Regularity on the Half-plane, Plane, and Strips
• Non Existence and Strong Ill-Posedness in $H^2$ for the Stable IPM Equation
• Finite time singularities of smooth solutions for the 2D incompressible porous media (IPM) equation with a smooth source