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

Mathematics > Analysis of PDEs [Submitted on 11 Mar 2026 (v1), last revised 18 Jun 2026 (this version, v3)] Title:Incompressible Euler Blowup at the $C^{1,\frac{1}{3}}$ Threshold View PDFAbstract: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) References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.