Singularity of the axisymmetric stagnation-point-like solution within a cylinder of the 3D Euler incompressible fluid equations

Abstract

In this paper we investigate analytically the formation of finite time singularities in the three dimensional incompressible Euler equations under the model of Gibbon, Fokas, and Doering for vorticity stretching within a bounded cylindrical domain and under axisymmetric conditions. We derive explicit Lagrangian solutions for the vorticity, its stretching rate, fluid pathlines, and velocity components by exploiting constants of motion associated with the field dependent infinitesimal symmetries of the system. The central finding is that the existence and nature of a finite time singularity are determined exclusively by the local geometric structure of the initial vortex stretching rate near its global minimum. Whether a singularity forms depends on how flat this profile is at the minimum. Flatter profiles delay the blowup and sufficient flatness can suppress it entirely. For power law behavior near the minimum, critical thresholds for the exponent are identified which separate regular solutions from those that develop a finite time singularity. These thresholds differ depending on whether the singularity occurs at the centre of the cylinder or on a ring away from the centre, with minima at the centre requiring higher flatness to avoid blowup. This work provides a rigorous analytical framework that elucidates how the local geometric structure of the initial conditions governs the potential for singularity formation in 3D fluid flows, offering fundamental insights into the interplay between symmetry, initial data, and the development of extreme events in idealised turbulence.

Figures (2)

• Figure 1: Evolution of the parabolic initial stretching rate $\gamma_0(r_0) = 20r_0^2 - 10$ (hence $f=-10,R=1$) and the uniform initial plane vorticity $\omega_0(r_0) = 10$ in the Eulerian description. The singularity time is computed as $T_* = \frac{\pi^2}{6|{\gminus}|} \approx 0.16449$. According to equation \ref{['eq:gamma<0']}, the asymptotic behavior of the stretching rate $\gamma$ is expected to take the form of a bell-shaped function, which is consistent with the spatial profile observed in this figure. While, at the center ($r=0$), $\gamma$ develops a finite-time singularity and $\omega$ develops a zero, at the boundary ($r=R$) $\omega$ exhibits a divergent peak, confirming their mutually exclusive blowup locations. The slower growth of $\omega$ compared to $\gamma$ (visible in the $z$-axis scale) aligns with its Lambert-W-type asymptotics derived in \ref{['eq:gamma<0']}.
• Figure 2: Evolution of the parabolic stretching rate $\gamma_0(r_0) = 10 - 20r_0^2$ (hence $f=10,R=1$) and the uniform initial plane vorticity $\omega_0(r_0) = 10$ in the Eulerian description. According to equation \ref{['eq:gamma>0']}, the asymptotic behavior of the stretching rate $\gamma$ is expected to exhibit a highly skewed profile near the boundary as the solution approaches the singularity time. This characteristic is reflected in the spatial distribution observed in the figure. In contrast to $\gamma$'s blowup being focused at the boundary, $\omega$ concentrates its blowup at the center ($r=0$), demonstrating the complementary nature of their singularities. The milder growth rate of $\omega$, evident from the $z$-axis scaling, is consistent with its Lambert-W-dominated asymptotics as predicted by \ref{['eq:gamma>0']}.

Theorems & Definitions (12)

• Remark 2.1.1
• Remark 2.2.1
• Remark 2.2.2
• Theorem 1: Pathline Solution and Lagrangian Velocity Field
• Remark 2.2.3
• Remark 2.2.4
• Remark 2.2.5
• Remark 2.2.6
• Remark 3.0.1
• Theorem 2