[Submitted on 11 Mar 2026 (v1), last revised 18 Jun 2026 (this version, v3)]

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Abstract:We prove finite-time Type--I blowup for the three-dimensional incompressible Euler equations in the axisymmetric no-swirl class, with initial velocity in $C^{1,\alpha}(\mathbb{R}^3)\cap L^2(\mathbb{R}^3)$, odd symmetry in $z$, and $0<\alpha<\tfrac13$, for an explicit class of finite-energy initial data. The singularity forms at a stagnation point on the symmetry axis. The axial strain and the global vorticity norm blow up at the Type--I rates $-\partial_z u_z(0,0,t)\simeq (T^*-t)^{-1}$ and $\|\omega(\cdot,t)\|_{L^\infty}\simeq (T^*-t)^{-1}$, while the meridional Jacobian collapses according to $J(t)\simeq (T^*-t)^{1/(1-3\alpha)}$. The proof is organized around a Lagrangian clock-and-driver framework. The clock is the meridional Jacobian $J(t)$, and the driver is the compressive axial strain $-\partial_z u_z(0,0,t)$. These variables satisfy, to leading order, a closed Riccati-clock system: the axial strain drives the collapse of $J(t)$, while the collapse of $J(t)$ amplifies the axial strain. We prove that the Euler flow tracks this clock-and-driver model up to the singular time. The main nonlocal obstruction is the pressure Hessian; it is controlled by a non-perturbative strain--pressure Hessian comparison showing that pressure cannot cancel the quadratic compressive strain responsible for collapse. This gives a dynamical explanation of the threshold $\alpha=\tfrac13$. The blowup mechanism is structurally stable and persists for an open set of admissible angular functions in a weighted Hölder topology.

Submission history

From: Steve Shkoller [view email]
[v1] Wed, 11 Mar 2026 16:33:06 UTC (116 KB)
[v2] Tue, 5 May 2026 16:38:16 UTC (201 KB)
[v3] Thu, 18 Jun 2026 15:14:36 UTC (164 KB)

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