Early-time resonances in the three-dimensional wall-bounded axisymmetric Euler and related equations

TL;DR

This paper analyzes the early-time resonances and singularity structures in solutions of 3D axisymmetric Euler equations using high-precision spectral methods and asymptotic analysis, revealing complex singularity distributions.

Contribution

It introduces a detailed spectral and asymptotic approach to identify and characterize early-time resonances and singularities in 3D Euler equations, linking these phenomena to tygers and spectral truncation effects.

Findings

Early-time resonances are observed in the solutions.

Singularities are distributed along two eye-shaped curves in the complex plane.

The results connect spectral resonances with solution singularities.

Abstract

We investigate the complex-time analytic structure of solutions of the 3D-axisymmetric, wall-bounded, incompressible Euler equations, by starting with the initial data proposed in Luo and Hou (2014), to study a possible finite-time singularity. We use our pseudospectral Fourier-Chebyshev method, with quadruple-precision arithmetic, to compute the time-Taylor series coefficients of the flow fields, up to a high order. We show that the resulting approximations display early-time resonances; the initial spatial location of these structures is different from that for the tygers, which we have obtained in Kolluru et al. (2022). We then perform asymptotic analysis of the Taylor-series coefficients, by using generalised ratio methods, to extract the location and nature of the convergence-limiting singularities and demonstrate that these singularities are distributed around the origin, in the complex-t2 plane, along two curves that resemble the shape of an eye. We obtain similar results for the 1D wall-approximation (of the full 3D-axisymmetric Euler equation) called the 1D HL model, for which we use Fourier-pseudospectral methods to compute the time-Taylor series coefficients of the flow fields. Our work examines the link between tygers, in Galerkin-truncated pseudospectral studies, and early-time resonances, in truncated time-Taylor expansions of solutions of PDEs, such as those we consider.