Incompressible Euler Blowup at the $C^{1,\frac{1}{3}}$ Threshold

TL;DR

The paper proves finite-time blowup for 3D incompressible Euler equations in an axisymmetric no-swirl setting with initial data less regular than $C^{1,1/3}$, using a novel Lagrangian framework and pressure analysis.

Contribution

It introduces a Lagrangian clock-and-strain method to establish blowup at the $C^{1,1/3}$ regularity threshold, advancing understanding of singularity formation.

Findings

Finite-time Type-I blowup occurs at the $C^{1,1/3}$ threshold.

On-axis axial strain and vorticity blow up at rate $(T^*-t)^{-1}$.

Collapse dynamics are governed by a Riccati law coupled with a clock ODE.

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 on-axis axial strain and the global vorticity norm blow up at the Type-I rates $-\partial_z u_z(0,0,t)\sim (T^*-t)^{-1}$ and $\|\omega(\cdot,t)\|_{L^\infty}\sim (T^*-t)^{-1}$, while the meridional Jacobian collapses according to $J(t)\sim (T^*-t)^{1/(1-3\alpha)}$. The proof introduces a Lagrangian clock-and-strain framework that replaces the Eulerian self-similar ansatz used in prior work with a Lagrangian flow decomposition. The collapse dynamics are governed by a Riccati law for the on-axis axial strain, coupled to a clock ODE for the meridional Jacobian. The decisive step is a non-perturbative strain-pressure comparison showing that the pressure Hessian 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 profiles in a weighted H\"older topology.