This paper derives two simple one‑space, one‑time (1+1)D systems from the three‑dimensional axisymmetric incompressible Euler equations and shows that their dynamics produce a finite‑time singularity at the coordinate origin (the apex). The key point is that these reduced systems are not toy models. They come from exact restrictions of a rigorously derived (1+2)D subsystem, so they capture a core singularity mechanism already present in the full axisymmetric Euler equations.

The author first rewrites the 3D axisymmetric Euler equations in a signed‑polar, pressure–velocity form on the meridian plane and isolates a (1+2)D subsystem called E2. By following the symmetry axes θ = 0 and θ = ±π/2 in polar coordinates (here θ is the angle on the meridian plane and x = √(r^2+z^2) is the signed radius), the dynamics along those axes close exactly. Those axis dynamics produce two rigorously derived (1+1)D systems, called (R0) for the horizontal axis and (Z0) for the vertical axis. The paper works with Hou–Li type variables {u,v,g} that the author calls building blocks of vorticity because their units and the structure of quadratic vortex stretching match the vorticity physics.

Tracing the axis systems to the apex x = 0 reduces the dynamics further to a single ordinary differential equation of Constantin–Lax–Majda (CLM) type. The CLM family is a well‑studied one‑dimensional model for vortex stretching that is known to produce finite‑time blow‑up under simple conditions. Using this reduction, the author proves that the apex trace of (R0) and (Z0) develops a genuine finite‑time singularity at the coordinate origin.

Why this matters: the question of whether smooth solutions of the 3D Euler equations can develop singularities in finite time is a long‑standing open problem in fluid dynamics. This work isolates an exact and transparent singularity mechanism that sits inside a faithful reduction of the 3D axisymmetric system. Because the reduced systems come directly from the full equations and preserve the divergence‑free and symmetry structure, the result strengthens the link between one‑dimensional blow‑up models and the full three‑dimensional problem.