Complex-time singular structure of the 1D Hou-Luo model

TL;DR

This paper analyzes the complex-time singularity structure of the 1D Hou-Luo model, revealing the nature of blow-up, singularity profiles, and validating criteria for finite-time singularities using a novel Lagrangian approach.

Contribution

It introduces a new Lagrangian formulation for the HL model, enabling detailed analysis of singularities and blow-up behavior in complex time, extending methods from Burgers equation studies.

Findings

Lagrangian series converge within a positive complex-time radius

Accurate prediction of blow-up time and singularity exponent

Development of a Lagrangian singularity theory explaining Eulerian singularity profiles

Abstract

Starting from smooth initial data, we investigate the complex-time analytic structure of the one-dimensional Hou--Luo (HL) model, a wall approximation of the three-dimensional axisymmetric Euler equations. While the finite-time blow-up in this setting has been already established, here we chart the entire singular landscape. This analysis is enabled by a novel formulation of the HL model in Lagrangian coordinates, in which the time-Taylor coefficients of the flow fields are evaluated symbolically to high truncation order. Our results are threefold. First, we show that the Lagrangian series for the vorticity converges within the complex-time disc of radius~$t_\star >0$ and is free from (early-time) resonances that impede the Eulerian formulation. Second, applying asymptotic analysis on the series, we recover both the blow-up time and the singularity exponent with high accuracy. This also enables a quantitative assessment of the Beale--Kato--Majda criterion, which we find correctly identifies the blow-up time, but washes out the local singularity exponent, as it relies on a spatial supremum. Third, and most importantly, we develop a Lagrangian singularity theory that predicts the eye-shaped singularity profile observed in Eulerian coordinates by exploiting the driving mechanism of the blow-up: The accumulation of multiple fluid particles at the same Eulerian position. The employed techniques extend recently introduced methods for the inviscid Burgers equation [C. Rampf et al., Phys. Rev. Fluids 7 (2022) 10, 104610], and can be further adapted to higher spatial dimensions or other hydrodynamical equations.