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Finite-Time Singularity Formation for $C^{1,α}$ Solutions to the Incompressible Euler Equations on $\mathbb{R}^3$

Tarek M. Elgindi

TL;DR

The paper develops a rigorous route to finite-time singularity formation for certain $C^{1,\alpha}$ (with small $\alpha$) solutions to the 3D incompressible Euler equations on $\mathbb{R}^3$. By reducing to axisymmetric no-swirl flows, reformulating in polar/self-similar coordinates, and isolating a fundamental nonlocal model, the authors construct explicit self-similar blow-up profiles and prove their stability under perturbations within carefully chosen weighted spaces. The approach blends nonlinear energy methods, spectral coercivity for linearized operators, elliptic regularity in weighted Sobolev-type spaces, and modulation analysis to control radiation and ensure the persistence of the blow-up in the full Euler dynamics. The results highlight a concrete mechanism for finite-time singularity formation in the $C^{1,\alpha}$ regime, clarify the role of nonlocal vortex-stretching terms, and provide a framework potentially extendable to more general settings and domains.

Abstract

It has been known since work of Lichtenstein [42] and Gunther [29] in the 1920's that the $3D$ incompressible Euler equation is locally well-posed in the class of velocity fields with Hölder continuous gradient and suitable decay at infinity. It is shown here that these local solutions can develop singularities in finite time, even for some of the simplest three-dimensional flows.

Finite-Time Singularity Formation for $C^{1,α}$ Solutions to the Incompressible Euler Equations on $\mathbb{R}^3$

TL;DR

The paper develops a rigorous route to finite-time singularity formation for certain (with small ) solutions to the 3D incompressible Euler equations on . By reducing to axisymmetric no-swirl flows, reformulating in polar/self-similar coordinates, and isolating a fundamental nonlocal model, the authors construct explicit self-similar blow-up profiles and prove their stability under perturbations within carefully chosen weighted spaces. The approach blends nonlinear energy methods, spectral coercivity for linearized operators, elliptic regularity in weighted Sobolev-type spaces, and modulation analysis to control radiation and ensure the persistence of the blow-up in the full Euler dynamics. The results highlight a concrete mechanism for finite-time singularity formation in the regime, clarify the role of nonlocal vortex-stretching terms, and provide a framework potentially extendable to more general settings and domains.

Abstract

It has been known since work of Lichtenstein [42] and Gunther [29] in the 1920's that the incompressible Euler equation is locally well-posed in the class of velocity fields with Hölder continuous gradient and suitable decay at infinity. It is shown here that these local solutions can develop singularities in finite time, even for some of the simplest three-dimensional flows.

Paper Structure

This paper contains 58 sections, 46 theorems, 527 equations.

Table of Contents

  1. Introduction
  2. The Euler equation
  3. The vorticity equation
  4. Statement of the Main Theorem
  5. Previous works on singularity formation
  6. Classical vs. Smooth and $\mathbb{R}^3$ vs. $\mathbb{R}^3_+$
  7. Organization
  8. Notation
  9. Functions, variables, and parameters
  10. Norms and Operators
  11. Linearized Operators
  12. The Setup
  13. Passing to a form of polar coordinates
  14. Reductions by taking $\alpha$ small and looking at $R=0$
  15. Dropping the transport term
  16. ...and 43 more sections

Key Result

Theorem 1

There exists an $\alpha>0$ and a divergence-free and odd $u_0\in C^{1,\alpha}(\mathbb{R}^3)$ with initial vorticity $|\omega_0(x)|\leq\frac{C}{|x|^\alpha+1}$ for some constant $C>0$ so that the unique local odd solution to E-IC (with $f\equiv 0$) belonging to the class $C^{1,\alpha}_{x,t}([0,1)\time

Theorems & Definitions (94)

  • Definition 1.2
  • Theorem 1
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 2.1
  • Remark 3.1
  • Lemma 4.1
  • Definition 5.1
  • Definition 5.2
  • ...and 84 more