A unified Boussinesq--Euler formulation and finite-time blow-up for a Hou--Luo type boundary-jet system
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913133500563456 |
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| author | Shi, Yaoming |
| author_facet | Shi, Yaoming |
| contents | 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 $Θ=\vartheta/r$ and write $Θ=u^2$ only when a smooth square-root branch has been fixed; equivalently, one may keep the scalar variable $Θ$ 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 $φ_{qq}(x,1,t)$. If the boundary jet is closed by the first-order Taylor truncation $φ_{qq}(x,1,t)=0$, it reduces to a closed unified $(1+1)$D system $(Q0)$ with the local boundary velocity law $u=-(m+2)^{-1}ω$. 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--Šverák--Yao. The theorem is therefore a blow-up result for the closed boundary-jet model, not for the unrestricted Boussinesq or Euler systems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_16322 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A unified Boussinesq--Euler formulation and finite-time blow-up for a Hou--Luo type boundary-jet system Shi, Yaoming Analysis of PDEs 35B44, 35B40, 35Q86, 76B03, 76D05 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 $Θ=\vartheta/r$ and write $Θ=u^2$ only when a smooth square-root branch has been fixed; equivalently, one may keep the scalar variable $Θ$ 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 $φ_{qq}(x,1,t)$. If the boundary jet is closed by the first-order Taylor truncation $φ_{qq}(x,1,t)=0$, it reduces to a closed unified $(1+1)$D system $(Q0)$ with the local boundary velocity law $u=-(m+2)^{-1}ω$. 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--Šverák--Yao. The theorem is therefore a blow-up result for the closed boundary-jet model, not for the unrestricted Boussinesq or Euler systems. |
| title | A unified Boussinesq--Euler formulation and finite-time blow-up for a Hou--Luo type boundary-jet system |
| topic | Analysis of PDEs 35B44, 35B40, 35Q86, 76B03, 76D05 |
| url | https://arxiv.org/abs/2605.16322 |