Non Existence and Strong Ill-Posedness in $H^2$ for the Stable IPM Equation

Abstract

We prove the non-existence and strong ill-posedness of the Incompressible Porous Media (IPM) equation for initial data that are small $H^2(\mathbb{R}^2)$ perturbations of the linearly stable profile $-x_2$. A remarkable novelty of the proof is the construction of an $H^2$ perturbation, which solves the IPM equation and neutralizes the stabilizing effect of the background profile near the origin, where a strong deformation leading to non-existence in $H^2$ is created. This strong deformation is achieved through an iterative procedure inspired by the work of C\'{o}rdoba and Mart\'{\i}nez-Zoroa (Adv. Math. 2022). However, several differences - beyond purely technical aspects - arise due to the anisotropic and, more importantly, to the partially dissipative nature of the equation, adding further challenges to the analysis.