Mild ill-posedness in $W^{1,\infty}$ for the incompressible porous media equation

Abstract: In this paper, we establish the mild ill-posedness of 2D IPM equation in the critical Sobolev space $W^{1,\infty}$ when the initial data are small perturbations of stable profile $g(x_2).$ Consequently, instability can be inferred. Notably, our results are valid for arbitrary vertically stratified density profiles $g(x_2)$ without imposing any restrictions on the sign of $g'(x_2).$ From a physical perspective, since gravity acts downward, density profiles satisfying $g'(x_2) < 0$ typically correspond to stable configurations, whereas those with $g '(x_2) > 0$ are generally expected to be unstable. Surprisingly, our analysis uncovers an unexpected instability even when $g'(x_2) < 0$ and $g'(x_2)\in W^{2,\infty}(\mathbb{R})$. To the best of our knowledge, this work provides the first rigorous demonstration of IPM instability for vertically nonlinear density profiles, marking a significant departure from conventional physical expectations.