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On the optimal rate of vortex stretching for axisymmetric Euler flows without swirl

Deokwoo Lim, In-Jee Jeong

TL;DR

This work determines the optimal upper bound on the growth of the azimuthal vorticity for axisymmetric Euler flows without swirl, showing $ abla hetaoldsymbol{ abla}^ heta abla heta abla abla$ grows at most like $(1+|t|)^{4/3}$ for smooth, compactly supported data. The authors develop a novel velocity bound that couples kinetic energy with conserved quantities tied to the vorticity, and they decompose the domain into axial and effectively two-dimensional regions to extract the sharp rate. The key technical advance is the $R^{1/4}$ estimate for the radial velocity at the maximal vorticity support, which, together with strategic estimates and conservation laws, yields the $t^{4/3}$ bound and clarifies the optimality via Childress’ eroding dipole model. The results extend to higher dimensions, producing dimension-dependent growth rates and highlighting the role of axisymmetry in constraining vorticity amplification, with implications for the global regularity and asymptotic behavior of axisymmetric flows.

Abstract

For axisymmetric flows without swirl and compactly supported initial vorticity, we prove the upper bound of $t^{4/3}$ for the growth of the vorticity maximum, which was conjectured by Childress [Phys. D, 2008] and supported by numerical computations from Childress--Gilbert--Valiant [J. Fluid Mech. 2016]. The key is to estimate the velocity maximum by the kinetic energy together with conserved quantities involving the vorticity.

On the optimal rate of vortex stretching for axisymmetric Euler flows without swirl

TL;DR

This work determines the optimal upper bound on the growth of the azimuthal vorticity for axisymmetric Euler flows without swirl, showing grows at most like for smooth, compactly supported data. The authors develop a novel velocity bound that couples kinetic energy with conserved quantities tied to the vorticity, and they decompose the domain into axial and effectively two-dimensional regions to extract the sharp rate. The key technical advance is the estimate for the radial velocity at the maximal vorticity support, which, together with strategic estimates and conservation laws, yields the bound and clarifies the optimality via Childress’ eroding dipole model. The results extend to higher dimensions, producing dimension-dependent growth rates and highlighting the role of axisymmetry in constraining vorticity amplification, with implications for the global regularity and asymptotic behavior of axisymmetric flows.

Abstract

For axisymmetric flows without swirl and compactly supported initial vorticity, we prove the upper bound of for the growth of the vorticity maximum, which was conjectured by Childress [Phys. D, 2008] and supported by numerical computations from Childress--Gilbert--Valiant [J. Fluid Mech. 2016]. The key is to estimate the velocity maximum by the kinetic energy together with conserved quantities involving the vorticity.
Paper Structure (23 sections, 8 theorems, 138 equations, 2 figures)

This paper contains 23 sections, 8 theorems, 138 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Wellposedness for axisymmetric Euler without swirl
  4. Optimal vorticity growth and eroding dipole model by Childress
  5. Velocity bounds for axisymmetric flows: the proof strategy
  6. Exponential upper bound
  7. Feng--Sverak estimate and the $t^{2}$ bound again
  8. Global velocity bound using kinetic energy and the $t^{3/2}$ bound
  9. Strategy and heuristic for the $t^{4/3}$ bound
  10. Growth rates in higher dimensions and related models
  11. Organization of the paper
  12. Preliminaries
  13. Notations
  14. Axisymmetric Biot--Savart law
  15. Proof of the main result
  16. ...and 8 more sections

Key Result

Theorem 1.1

Assume that $\omega^{\theta}_{0}$ is compactly supported and $\Vert r^{-1} \omega^{\theta}_{0} \Vert_{L^\infty(\mathbb R^{3})}$ is bounded. Then the corresponding unique global-in-time solution $\omega(t,\cdot)$ of eq:Vorticityform belonging to $L^{\infty}_{loc}(\mathbb R;L^{\infty}(\mathbb R^3))$ w

Figures (2)

  • Figure 1: Vortex dipole with maximal support radius $R$ and length scale $L$.
  • Figure 2: A diagram showing of $(I_{1})$ from \ref{['eq:urRzsplit']} and $(I_{21})$ from \ref{['eq:I2split']} when $l<1$, with $z = 0$. The darkest region represents $(I_{22})$, which is $(I_{2})$ minus $(I_{21})$.

Theorems & Definitions (18)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1: FeSv
  • Remark 2.2
  • Proposition 3.1
  • proof
  • proof : Proof of Theorem \ref{['thm:growth-optimal']}
  • Lemma 3.2
  • ...and 8 more