2D inviscid Boussinesq equations and 3D axisymmetric Euler equations: (1) A unification ($Em$), (2) Finite-time blow-up of two unified $(1+1)$D systems rigorously derived from ($Em$)

Abstract

We derive a unified polar $(1+2)$D subsystem $(Em)$, with $m=1,2$, from the 2D inviscid Boussinesq and 3D axisymmetric Euler equations. On the symmetry axes $\theta=0,\pm\pi/2,\pi$, ridge flatness closes the dynamics and gives two exact unified $(1+1)$D reductions: the horizontal-axis system $(R0)$ and the vertical-axis system $(Z0)$. Their common apex trace is a Constantin--Lax--Majda type ODE that yields finite-time blow-up at $x=0$. Subsection~\ref{seq:vorticity-strain} connects this pointwise mechanism with Euler continuation theory: for any compatible axisymmetric realization, explicit apex blow-up forces divergence of the time-integrated $L^\infty$ norm of $\nabla\boldsymbol v$, so the singularity is detected by the strain criterion. Section~\ref{sec:R0-SS} strengthens the reduced mechanism by constructing regular apex-only self-similar profiles for the convective horizontal-axis equation $(R0)$; the resulting solution is bounded away from $x=0$, blows up at the apex, and satisfies the same strain-divergence condition. Finally, we derive the exact background--remainder equations and state a conditional nonlinear stability framework: if a compatible full background, weighted elliptic/coercive estimates, and a spectral gap exponent are available, then the apex blow-up transfers to the full solution. Thus the rigorous components are the derivation of $(Em)$, $(R0)$, and $(Z0)$, the apex blow-up and strain verification, the apex-only $(R0)$ self-similar construction, and the perturbative framework; the remaining open step is the unconditional construction and control of the full background away from the apex.