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 $\partial_X u(t,0,Z)$ 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.