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Description

We construct an odd smooth initial density for the unforced incompressible porous media (IPM) equation on $\mathbb{T} ^2$ whose solution develops a finite-time blowup. The components $\partial_{x_2}\rho(t,0)$ and $\partial_{x_1}u_1(t,0)$ tend to $+\infty$ as the maximal smooth existence time is approached, while the density and velocity remain bounded in $L^2$. Starting from a stationary solution, we iterate the angular amplification of localized oscillations with rapidly growing frequencies,  adapting the mechanism of C\'ordoba and Mart\'inez-Zoroa~\cite{CMZ} for forced IPM. Each perturbation is constructed from time zero and creates the geometry configuration for the next iteration stage. A one-sided upper bound for the pressure Hessian is applied to control backward preparation and initial perturbations, following an idea from the Euler construction of OpenAI.