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Vorticity blow-up for the 3D incompressible Euler equations

Wenjie Deng, Song Jiang, Minling Li, Zhaonan Luo

TL;DR

The paper proves finite-time vorticity blow-up for the 3D incompressible Euler equations in the axisymmetric no-swirl setting by developing a refined self-similar framework. It introduces a scaling parameter $\beta$ and constructs fundamental self-similar profiles $F^{*}_{\gamma}$, establishing their parameter stability and embedding them into a critical-regularity energy setting. By decomposing the solution into a main self-similar part plus a perturbation $g$ and proving coercivity and null-structure–driven energy estimates in $\mathcal{H}^2\cap\mathcal{E}^2$, the authors close the nonlinear energy estimates and demonstrate finite-time blow-up with a scaling index $\beta/\alpha$, along with precise time-integrability properties for the velocity components. The work provides a rigorous mechanism for singularity formation in Euler flows and delivers a robust analytic toolkit—weighted elliptic/transport estimates and coercivity in critical spaces—that may inform broader studies of self-similar blow-up and integrability near singularities.

Abstract

In this paper, we study the finite-time blow-up for classical solutions of the 3D incompressible Euler equations with low-regularity initial vorticity. Applying the self-similar method and stability analysis of the self-similar system in critical Sobolev space, we prove that the vorticity of the axi-symmetric 3D Euler equations develops a finite-time singularity with certain scaling indices. Furthermore, we investigate the time integrability of the solutions. The proof is based on the new observations for the null structure of the transport term, and the parameter stability of the fundamental self-similar models.

Vorticity blow-up for the 3D incompressible Euler equations

TL;DR

The paper proves finite-time vorticity blow-up for the 3D incompressible Euler equations in the axisymmetric no-swirl setting by developing a refined self-similar framework. It introduces a scaling parameter and constructs fundamental self-similar profiles , establishing their parameter stability and embedding them into a critical-regularity energy setting. By decomposing the solution into a main self-similar part plus a perturbation and proving coercivity and null-structure–driven energy estimates in , the authors close the nonlinear energy estimates and demonstrate finite-time blow-up with a scaling index , along with precise time-integrability properties for the velocity components. The work provides a rigorous mechanism for singularity formation in Euler flows and delivers a robust analytic toolkit—weighted elliptic/transport estimates and coercivity in critical spaces—that may inform broader studies of self-similar blow-up and integrability near singularities.

Abstract

In this paper, we study the finite-time blow-up for classical solutions of the 3D incompressible Euler equations with low-regularity initial vorticity. Applying the self-similar method and stability analysis of the self-similar system in critical Sobolev space, we prove that the vorticity of the axi-symmetric 3D Euler equations develops a finite-time singularity with certain scaling indices. Furthermore, we investigate the time integrability of the solutions. The proof is based on the new observations for the null structure of the transport term, and the parameter stability of the fundamental self-similar models.

Paper Structure

This paper contains 33 sections, 42 theorems, 778 equations.

Table of Contents

  1. Introduction
  2. Statement of main theorem
  3. Reformulation
  4. Motivation and main ideas
  5. Motivation
  6. Main ideas
  7. Conclusion
  8. Notations
  9. The fundamental model
  10. Blow-up phenomenon of the fundamental model
  11. Parameter stability and energy estimate of the fundamental self-similar solutions
  12. Self-similar variables and asymptotic analysis
  13. Decomposition of potential equation
  14. Asymptotic analysis of stretching term
  15. The null structure of transport term
  16. ...and 18 more sections

Key Result

Theorem 1.1

For any $\beta\in(0,1]$, there exists a positive constant $\alpha(\beta)$, the initial vorticity $\omega_0 \in C^{0,\frac{\alpha}{20\beta}}(D_0)$ with $|\omega_0|\leq c|\bm{x}|^{\frac{\alpha}{2\beta}}(1+|\bm{x}|)^{-\left({\frac{\alpha}{2\beta}+\frac{\alpha}{2}}\right)}$ for some constant $c>0$, and Furthermore, for any $1\leq p<\frac{2}{2-\beta}$, the velocity components exhibit the following int

Theorems & Definitions (86)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • ...and 76 more