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

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 $0<\alpha <\frac{1}{3}$, 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 $-{\partial }_z{u}_z(0,0,t)\simeq (T^{\ast }-t{)}^{-1}$ and $\Vert \omega (\cdot ,t){\Vert }_{L^{\infty }}\simeq (T^{\ast }-t{)}^{-1}$, while the meridional Jacobian collapses according to $J(t)\simeq (T^{\ast }-t{)}^{1/(1-3\alpha )}$. 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 $\alpha =\frac{1}{3}$. The blowup mechanism is structurally stable and persists for an open set of admissible angular functions in a weighted Hölder topology.

arXiv comment159 pages; simplified the proof of the pressure Hessian bounds and improved the exposition