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On putative self-similarity for incompressible 3D Euler

Peter Constantin, Mihaela Ignatova, Vlad Vicol

TL;DR

This work analyzes hypothetical finite-time self-similar blowups of the incompressible $3$D Euler equations without boundaries. It establishes rigorous lower bounds on the similarity exponent $\gamma$ under finite kinetic energy and in axisymmetric settings, and connects $\gamma$ to the behavior of the velocity field away from the singular core. Core tools include a vorticity-stretching decomposition, the self-similar Lagrangian flow with Cauchy/Weber representations, Kelvin circulation, and the Bernoulli function, which together reveal $\gamma=1/2$ as a distinguished threshold under an outgoing property. Consequently, an outgoing globally self-similar Euler solution cannot realize $\gamma<1/2$, and axisymmetric configurations further enforce $\gamma\ge 1/2$, constraining the route to Navier–Stokes singularities via Euler self-similarity.

Abstract

We consider hypothetical solutions of 3D Euler which blow up in finite time in a self-similar fashion. We prove that if the initial data has finite kinetic energy, then the similarity exponent $γ$ which governs the rate of zooming in must be larger than $2/5$. If a smooth globally self-similar blowup profile exists, and this profile satisfies an outgoing property, we prove that $γ\geq 1/2$. For axisymmetric solutions, we establish the bound $γ\geq 1/2$ in more general settings, including ones in which the outgoing property is not present.

On putative self-similarity for incompressible 3D Euler

TL;DR

This work analyzes hypothetical finite-time self-similar blowups of the incompressible D Euler equations without boundaries. It establishes rigorous lower bounds on the similarity exponent under finite kinetic energy and in axisymmetric settings, and connects to the behavior of the velocity field away from the singular core. Core tools include a vorticity-stretching decomposition, the self-similar Lagrangian flow with Cauchy/Weber representations, Kelvin circulation, and the Bernoulli function, which together reveal as a distinguished threshold under an outgoing property. Consequently, an outgoing globally self-similar Euler solution cannot realize , and axisymmetric configurations further enforce , constraining the route to Navier–Stokes singularities via Euler self-similarity.

Abstract

We consider hypothetical solutions of 3D Euler which blow up in finite time in a self-similar fashion. We prove that if the initial data has finite kinetic energy, then the similarity exponent which governs the rate of zooming in must be larger than . If a smooth globally self-similar blowup profile exists, and this profile satisfies an outgoing property, we prove that . For axisymmetric solutions, we establish the bound in more general settings, including ones in which the outgoing property is not present.
Paper Structure (19 sections, 12 theorems, 93 equations)

This paper contains 19 sections, 12 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. On asymptotically self-similar blowup with finite kinetic energy
  3. On globally self-similar blowup
  4. The self-similar ansatz
  5. The stationary self-similar PDE
  6. Space rescaling and normalization
  7. Behavior of the profiles at space infinity
  8. A remark for the viscous case.
  9. Self-similar vortex stretching
  10. Self-similar Lagrangian particle trajectories
  11. The self-similar Cauchy formula and the Weber formula
  12. The self-similar Kelvin circulation theorem
  13. The self-similar Bernoulli function
  14. An outgoing property?
  15. Axisymmetric global self-similarity
  16. ...and 4 more sections

Key Result

Theorem 2.1

Assume the 3D Euler equation eq:Euler has initial data $\omega_0 \in C^1(\mathbb{R}^3)$ with finite kinetic energy, i.e. $u_0 = (-\Delta)^{-1} \nabla \times \omega_0 \in L^2(\mathbb{R}^3)$. Assume that the resulting local-in-time smooth solution blows up at some finite time $T_*>0$. If there exists then $\gamma \geq 2/5$.

Theorems & Definitions (29)

  • Theorem 2.1
  • Remark 2.2
  • proof : Proof of Theorem \ref{['thm:two:fifths']}
  • Proposition 3.1
  • Remark 3.2
  • proof : Proof of Proposition \ref{['nse']}
  • Proposition 3.3
  • Theorem 3.4
  • proof : Proof of Theorem \ref{['thm:Omega:not:small']}
  • Remark 3.5
  • ...and 19 more