A unified Boussinesq--Euler formulation and finite-time blow-up for a Hou--Luo type boundary-jet system

TL;DR

This paper introduces a unified formulation for inviscid fluid systems in the meridian plane, derives a boundary jet model, and proves finite-time blow-up for a simplified boundary-jet system using a Riccati argument.

Contribution

It develops a unified vorticity-stream formulation for Boussinesq and Euler systems and establishes finite-time blow-up for a boundary-jet model.

Findings

Finite-time blow-up proven for the boundary-jet model.

Unified formulation encodes 2D Boussinesq and 3D Euler equations.

Boundary jet closure leads to a solvable 1+1D system.

Abstract

We derive a unified vorticity--stream formulation $(Bm)$ for two parity-reduced inviscid systems in the meridian plane: the 2D inviscid Boussinesq equations $(m=1)$ and the 3D axisymmetric Euler equations with swirl $(m=2)$. In the Boussinesq case we set $\Theta=\vartheta/r$ and write $\Theta=u^2$ only when a smooth square-root branch has been fixed; equivalently, one may keep the scalar variable $\Theta$ throughout. In the squared radial variable $q=r^2$, the two cases are encoded by the same parameterized system with $m=1,2$. At the boundary $q=1$, a Taylor expansion gives an exact boundary jet: the transport equations close on the boundary, while the elliptic relation also contains the next normal jet $\varphi_{qq}(x,1,t)$. If the boundary jet is closed by the first-order Taylor truncation $\varphi_{qq}(x,1,t)=0$, it reduces to a closed unified $(1+1)$D system $(Q0)$ with the local boundary velocity law $u=-(m+2)^{-1}\omega$. We prove finite-time blow-up for this closed Hou--Luo type model on a periodic interval by a Riccati argument in the spirit of Choi--Hou--Kiselev--Luo--\v{S}ver\'ak--Yao. The theorem is therefore a blow-up result for the closed boundary-jet model, not for the unrestricted Boussinesq or Euler systems.