Near-blowup asymptotic analysis and analytical extension for the inviscid primitive equations
Abstract
The primitive equations widely approximate the Navier--Stokes equations in large-scale geophysical fluid dynamics. Their finite-time singularity formation-which occurs exclusively in the inviscid case-sheds light on the mechanisms driving small-scale creation. Building on \cite{Collot2023}, which constructed 2D horizontally odd solutions where the on-axis derivative ∂X u(t,0,Z) blows up at rate (T−t)−1 with boundary concentration scale |ln (T−t)|−1, this paper characterizes this blow-up as forming between hyperbolic stagnation points. We derive precise asymptotic expansions for on-axis derivatives beyond leading order, identifying a horizontal concentration scale of (T−t)3/2 characteristic of generic Burgers' singularities. Furthermore, we justify the 2D solution's expansion near the symmetry axis by proving (T−t)3/2 is a universal lower bound on the radius of analyticity.
Related
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BibTeX
@misc{collot2026nearblowup,
title = {{Near-blowup asymptotic analysis and analytical extension for the inviscid primitive equations}},
author = {Charles Collot and Slim Ibrahim and Lingjun Qian},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28403},
eprint = {2610.28403},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28403},
}
RIS
TY - UNPB
TI - Near-blowup asymptotic analysis and analytical extension for the inviscid primitive equations
AU - Charles Collot
AU - Slim Ibrahim
AU - Lingjun Qian
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28403
UR - https://arxiv.org/abs/2610.28403
AB - The primitive equations widely approximate the Navier--Stokes equations in large-scale geophysical fluid dynamics. Their finite-time singularity formation-which occurs exclusively in the inviscid case-sheds light on the mechanisms driving small-scale creation. Building on \cite{Collot2023}, which constructed 2D horizontally odd solutions where the on-axis derivative ∂_X u(t,0,Z) blows up at rate (T−t)^−1 with boundary concentration scale |ln (T−t)|^−1, this paper characterizes this blow-up as forming between hyperbolic stagnation points. We derive precise asymptotic expansions for on-axis derivatives beyond leading order, identifying a horizontal concentration scale of (T−t)^3/2 characteristic of generic Burgers' singularities. Furthermore, we justify the 2D solution's expansion near the symmetry axis by proving (T−t)^3/2 is a universal lower bound on the radius of analyticity.
ER -