Finite-time blowup for 2D unforced IPM
Abstract
We construct an odd smooth initial density for the unforced incompressible porous media (IPM) equation on 𝕋2 whose solution develops a finite-time blowup. The components ∂x2ρ(t,0) and ∂x1u1(t,0) tend to +∞ as the maximal smooth existence time is approached, while the density and velocity remain bounded in L2. 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].
Related
- The Positive Defect Problem: Target and Admissibility Criteria for a Programmatic Search for Unforced Navier-Stokes Blowup
- Finite-time blowup of the Euler equation without forcing
- Finite-Time Blowup for Navier-Stokes with Smooth Forcing, Part II: Residual Correction via Oscillatory Pulses
- Finite-Time Blowup for Navier-Stokes with Smooth Forcing, Part I: Construction of Self-Similar Solutions with Admissible Stress and Flat Remainder
- Finite-time blowup of the IPM equation from smooth initial data without forcing
BibTeX
@misc{dai2026finitetime,
title = {{Finite-time blowup for 2D unforced IPM}},
author = {Mimi Dai},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06712},
eprint = {2610.06712},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06712},
}
RIS
TY - UNPB
TI - Finite-time blowup for 2D unforced IPM
AU - Mimi Dai
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06712
UR - https://arxiv.org/abs/2610.06712
AB - We construct an odd smooth initial density for the unforced incompressible porous media (IPM) equation on 𝕋^2 whose solution develops a finite-time blowup. The components ∂_x_2ρ(t,0) and ∂_x_1u_1(t,0) tend to +∞ 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].
ER -