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 ($Em$)

Yaoming Shi

paper

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

arXiv:2603.26715

Abstract

We derive a unified polar $(1 + 2)$ D subsystem $(E m)$ , with $m = 1, 2$ , from the 2D inviscid Boussinesq and 3D axisymmetric Euler equations. On the symmetry axes $θ = 0, \pm π /2, π$ , ridge flatness closes the dynamics and gives two exact unified $(1 + 1)$ D reductions: the horizontal-axis system $(R 0)$ and the vertical-axis system $(Z 0)$ . 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 $\nabla v$ , 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 $(R 0)$ ; 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 $(E m)$ , $(R 0)$ , and $(Z 0)$ , the apex blow-up and strain verification, the apex-only $(R 0)$ 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.

arXiv commentThis revision adds a vorticity-strain verification of finite-time blow-up and constructs an explicit apex-only self-similar (R0) profile, connecting the reduced Hou-Li axis dynamics to the standard Euler strain continuation criterion

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

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