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

In this paper, we establish the mild ill-posedness of 2D IPM equation in the critical Sobolev space $W^{1,\mathrm{\infty }}$ when the initial data are small perturbations of stable profile $g(x_2)\mathrm{.}$ 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 $g^{\mathrm{\prime }}(x_2)\mathrm{.}$ From a physical perspective, since gravity acts downward, density profiles satisfying $g^{\mathrm{\prime }}(x_2)<0$ typically correspond to stable configurations, whereas those with $g^{\mathrm{\prime }}(x_2)>0$ are generally expected to be unstable. Surprisingly, our analysis uncovers an unexpected instability even when $g^{\mathrm{\prime }}(x_2)<0$ and $g^{\mathrm{\prime }}(x_2)\in W^{2,\mathrm{\infty }}(\mathbb{R})$. 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.