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Extending the Córdoba-Martínez-Zoroa IPM Blow-Up to Uniformly Space-Time Smooth Forcing
Abstract
By adapting the techniques used in the IPM blowup result of Córdoba-Martínez-Zoroa with spatially smooth force, we prove finite-time blow-up for the IPM equation on 𝕋2 with a uniformly spacetime smooth force. In particular, we show there exist a smooth odd initial density, a smooth odd force F∈ C∞([0,1]×𝕋2), and a classical solution ρ on [0,1) whose density gradient and spatial velocity gradient diverge in L∞ as t↑ 1. Nevertheless, ρ(t) converges in Cη for every 0≤η<1.
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BibTeX
@misc{alpoge2026extending,
title = {{Extending the Córdoba-Martínez-Zoroa IPM Blow-Up to Uniformly Space-Time Smooth Forcing}},
author = {Levent Alpöge and Tristan Buckmaster and Matei P. Coiculescu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16470},
eprint = {2609.16470},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16470},
}
RIS
TY - UNPB
TI - Extending the Córdoba-Martínez-Zoroa IPM Blow-Up to Uniformly Space-Time Smooth Forcing
AU - Levent Alpöge
AU - Tristan Buckmaster
AU - Matei P. Coiculescu
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16470
UR - https://arxiv.org/abs/2609.16470
AB - By adapting the techniques used in the IPM blowup result of Córdoba-Martínez-Zoroa with spatially smooth force, we prove finite-time blow-up for the IPM equation on 𝕋^2 with a uniformly spacetime smooth force. In particular, we show there exist a smooth odd initial density, a smooth odd force F∈ C^∞([0,1]×𝕋^2), and a classical solution ρ on [0,1) whose density gradient and spatial velocity gradient diverge in L^∞ as t↑ 1. Nevertheless, ρ(t) converges in C^η for every 0≤η<1.
ER -