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

Steve Shkoller

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 $C^{1,α}(\R^3)\cap L^2(\R^3)$ and odd symmetry in $z$, for \emph{every} $α\in(0,\tfrac13)$. Since axisymmetric no-swirl solutions with $C^{1,α}$ velocity are globally regular for $α>\tfrac13$, 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 $\|\bsω(\cdot,t)\|_{L^\infty}\sim(T^*-t)^{-1}$, $-\partial_z u_z(0,0,t)\sim(T^*-t)^{-1}$, and the meridional Jacobian collapsing as $J(t)\sim\big(Γ(T^*-t)\big)^{1/(1-3α)}$. 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.

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 and odd symmetry in , for \emph{every} . Since axisymmetric no-swirl solutions with velocity are globally regular for , this result is sharp up to the endpoint: it covers the entire open interval , 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 , , and the meridional Jacobian collapsing as . 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 . The blowup mechanism is structurally stable: it persists for an open set of admissible angular profiles in a weighted topology.
Paper Structure (109 sections, 54 theorems, 832 equations)

This paper contains 109 sections, 54 theorems, 832 equations.

Table of Contents

  1. Introduction
  2. Axisymmetric no-swirl and rough velocities
  3. The $C^{1,\alpha}$ threshold in axisymmetry
  4. A brief landscape of rigorous blowup results
  5. Summary of the gap
  6. Key new ideas
  7. Main results
  8. Why an Eulerian self-similar ansatz breaks down near $\alpha=\tfrac{1}{3}$
  9. Methodology: a Lagrangian strain analysis
  10. The Axisymmetric No-Swirl Setting
  11. Cylindrical basis vectors
  12. The velocity gradient tensor
  13. The pressure equation source term $S$
  14. Lagrangian Variables and Vorticity Transport
  15. The flow map
  16. ...and 94 more sections

Key Result

Theorem 1.1

We fix $\alpha\in(0,\tfrac{1}{3})$. There exist $\gamma>\alpha+\tfrac{5}{2}$ and a constant such that the following holds. For any $0<\Gamma\le\Gamma_0$, let $u_0^*$ be the axisymmetric no-swirl initial datum, odd in $z$, whose angular vorticity is given by eq:vort0 with perturbation $h\equiv 0$ (equivalently, the distinguished zero-perturbation datum generated by the Target Profile $\Thet If de

Theorems & Definitions (141)

  • Theorem 1.1: Finite-time Type--I blowup for the Target Profile
  • Theorem 1.2: Open-set stability of the Target-Profile blowup
  • Lemma 4.1: Axial Flow Map Properties
  • proof : Proof of Lemma \ref{['lem:flow1']}
  • Lemma 4.2
  • proof : Proof of Lemma \ref{['lem2']}
  • Remark 5.1: A Type--I criterion in terms of the pressure Hessian (Chae--Constantin)
  • Definition 5.2: Admissible Initial Data Class $A_{\alpha,\gamma}$
  • Lemma 5.3: Regularity and finite energy
  • Remark 5.4: The regularity hypothesis in Theorem \ref{['thm:main']}
  • ...and 131 more