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 $$C^{{1,α}(\R^3)\cap} L<sup>2(\R<sup>3)$$ and odd symmetry in $z$, for \emph{every} $α\in(0,\tfrac13)$. Since axisymmetric no-swirl solutions with $C<sup>{1,α}$ velocity are globally regular for $α&gt;\tfrac13$, this result is sharp up to the endpoint: it covers the entire open interval $(0, \tfrac{1}{3})$, reaching the structural regularity threshold from below. The singularity forms at the stagnation point on the symmetry axis, with vorticity and strain blowing up at the Type--I rate $|\bsω(\cdot,t)|_{L<sup>\infty}\sim(T<sup>*-t)<sup>{-1}$, $-\partial_z u_z(0,0,t)\sim(T<sup>*-t)<sup>{-1}$, and the meridional Jacobian collapsing as $J(t)\sim\big(Γ(T<sup>*-t)\big)<sup>{1/(1-3α)}$. The proof introduces a Lagrangian clock-and-driver framework that replaces the Eulerian self-similar ansatz used in prior work. The collapse dynamics are governed by a Riccati-type ODE for the axial strain, and the decisive step is a non-perturbative bound on the strain--pressure competition, established via a spectral decomposition of the angular pressure source, showing that the quadratic strain term dominates the resistive pressure Hessian uniformly for all $α\in(0,\tfrac13)$. The blowup mechanism is structurally stable: it persists for an open set of admissible angular profiles in a weighted topology.