Finite-time blowup for 2D unforced IPM

Abstract

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 Córdoba and Martínez-Zoroa [7] 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 [14].

arXiv comment51 pages