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Finite-time blow-up and conditional perturbative control for a $(1+2)$D system (E2) derived from the 3D axisymmetric Euler equations

Yaoming Shi

Abstract

In polar variables on the meridian plane, we study a closed $(1+2)$D system (E2) derived from the three-dimensional axisymmetric Euler equations under a parity ansatz. A central feature of the paper is the velocity--pressure formulation: it keeps the divergence-free structure visible, reveals the distinguished ridge rays, and leads to an exact apex-dynamics reduction on those rays. The reduced ridge system is a convection-free $(1+1)$D reaction system of Constantin--Lax--Majda type, which yields finite-time blow-up at the ridge apex. The paper has three main outputs. First, we derive system (E2) from the 3D axisymmetric Euler equations in Hou--Li type variables and identify the ridge rays on which the dynamics reduce to the CLM-type reaction system. Second, we derive the exact background--remainder equations in the $(x,ξ)$ variables and prove singular weighted linear estimates for the remainder system. Third, we formulate a conditional nonlinear control principle in the spirit of Elgindi--Jeong: if a compatible background exists on $[0,T)$ with the coefficient bounds required by the weighted energy method, and if the remainder stays subordinate to the background singularity in the detecting norm, then the full solution inherits the same finite-time blow-up. Our approach is complementary to vorticity--stream and boundary-driven singularity frameworks in the recent literature. Here the analysis is carried out on the full reduced-plane geometry attached to the pressure--velocity reduction, with smooth functions and with symmetry replacing boundary/irregularity mechanisms in the handling of convection. What is unconditional in the present paper is the exact reduction from 3D axisymmetric Euler, the exact ridge/apex blow-up dynamics, the apex flatness criterion at $x=0$, and the weighted remainder framework.

Finite-time blow-up and conditional perturbative control for a $(1+2)$D system (E2) derived from the 3D axisymmetric Euler equations

Abstract

In polar variables on the meridian plane, we study a closed D system (E2) derived from the three-dimensional axisymmetric Euler equations under a parity ansatz. A central feature of the paper is the velocity--pressure formulation: it keeps the divergence-free structure visible, reveals the distinguished ridge rays, and leads to an exact apex-dynamics reduction on those rays. The reduced ridge system is a convection-free D reaction system of Constantin--Lax--Majda type, which yields finite-time blow-up at the ridge apex. The paper has three main outputs. First, we derive system (E2) from the 3D axisymmetric Euler equations in Hou--Li type variables and identify the ridge rays on which the dynamics reduce to the CLM-type reaction system. Second, we derive the exact background--remainder equations in the variables and prove singular weighted linear estimates for the remainder system. Third, we formulate a conditional nonlinear control principle in the spirit of Elgindi--Jeong: if a compatible background exists on with the coefficient bounds required by the weighted energy method, and if the remainder stays subordinate to the background singularity in the detecting norm, then the full solution inherits the same finite-time blow-up. Our approach is complementary to vorticity--stream and boundary-driven singularity frameworks in the recent literature. Here the analysis is carried out on the full reduced-plane geometry attached to the pressure--velocity reduction, with smooth functions and with symmetry replacing boundary/irregularity mechanisms in the handling of convection. What is unconditional in the present paper is the exact reduction from 3D axisymmetric Euler, the exact ridge/apex blow-up dynamics, the apex flatness criterion at , and the weighted remainder framework.

Paper Structure

This paper contains 38 sections, 18 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. Related work and context.
  3. The derivation of system (E2) from 3D axisymmetric Euler equations
  4. Velocity--pressure formulation and Hou--Li type variables
  5. System (E$2$)
  6. Polar coordinates $(x=\sqrt{r^2+z^2},\ \theta=\arctan(z/r))$
  7. Background
  8. Ridge ray and Ridge functions
  9. Ridge blow-up and the full-wedge extension problem
  10. Perturbation PDEs
  11. Getting rid of $(p_\xi,p_x)$
  12. Explicit background, perturbation PDEs, and energy bounds
  13. Initial energy and finiteness
  14. Updated Linear PDEs and Nonlinear Terms
  15. Coefficient Functions
  16. ...and 23 more sections

Key Result

Theorem 2.4

Suppose $u_0(x)=u(0,x)$ is a smooth function that decays sufficiently rapidly as $|x|\to\infty$, and let $v_0(x)=v(0,x)=H(u_0)(x)$. Then the solution to the model vorticity system eq:CLM2 is explicitly given by $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (46)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4: Constantin--Lax--Majda explicit formula
  • Theorem 2.5: Constantin--Lax--Majda breakdown criterion
  • Remark 2.6: $t$-scaling factor ${\color{blue}{\lambda}}$ and $z$-scaling factor ${\mu}$
  • Remark 2.7
  • Theorem 2.8
  • proof : Proof of \ref{['thm:ridge-ray-ridge-functions']}
  • Theorem 2.9: Blow-up set criterion for CLM-$q$
  • ...and 36 more