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arXiv · 2610.06544

Finite-time blowup of the IPM equation from smooth initial data without forcing

Fan Zheng
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Abstract

We prove finite-time gradient blow-up for the unforced incompressible porous media equation on the two-dimensional torus, removing the forcing used in the IPM blow-up constructions of Alpöge, Buckmaster, and Coiculescu \cite{lit:ABC} and Córdoba and Martínez-Zoroa \cite{lit:CMIPM}. The key is an identity that expresses the logarithmic growth of a layer's gradient as the negative pressure curvature transverse to its transported phase. A nearly vertical parent layer rotates and amplifies a new phase, with a gain exponential in the reciprocal parent tilt. An asymmetric periodic profile gives a uniform upper bound for the pressure Hessian, preventing rapid decay of the gradient magnitude. Each layer is seeded at time zero; its phase means are selected by a terminal displacement problem, and a finite order expansion is corrected to an exact unforced solution. The scale induction then shows blow-up of two derivative components at the origin and convergence to a terminal density in every $C^η$, $η<1$.

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Fan Zheng. 2026-10-05. Finite-time blowup of the IPM equation from smooth initial data without forcing. https://arxiv.org/abs/2610.06544

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