Finite-time blowup of the IPM equation from smooth initial data without forcing
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 Cη, η<1.
Related
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BibTeX
@misc{zheng2026finitetime,
title = {{Finite-time blowup of the IPM equation from smooth initial data without forcing}},
author = {Fan Zheng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06544},
eprint = {2610.06544},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06544},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Finite-time blowup of the IPM equation from smooth initial data without forcing
AU - Fan Zheng
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06544
UR - https://arxiv.org/abs/2610.06544
AB - 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.
ER -