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arXiv · 2605.16322

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

Yaoming Shi
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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.

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Yaoming Shi. 2026-05-05. A unified Boussinesq--Euler formulation and finite-time blow-up for a Hou--Luo type boundary-jet system. https://arxiv.org/abs/2605.16322

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