A unified Boussinesq--Euler formulation and finite-time blow-up for a Hou--Luo type boundary-jet 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 ${\phi }_{qq}(x,1,t)$. If the boundary jet is closed by the first-order Taylor truncation ${\phi }_{qq}(x,1,t)=0$, 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--Šverák--Yao. The theorem is therefore a blow-up result for the closed boundary-jet model, not for the unrestricted Boussinesq or Euler systems.

13 pages. Unified Boussinesq--Euler formulation and finite-time blow-up for a closed Hou--Luo %type boundary-jet system