Symmetry-plane model of 3D Euler flows and mapping to regular systems to improve blowup assessment using numerical and analytical solutions

TL;DR

This paper introduces a family of 3D Euler flow models on a symmetry plane, demonstrating how a nonlinear mapping improves numerical blowup detection and accuracy, with implications for understanding finite-time singularities.

Contribution

A novel one-parameter family of models respecting the Euler equations' structure, with a mapped regular system that enhances numerical blowup analysis and accuracy.

Findings

Mapped system improves supremum norm and singularity time estimates

Analyticity-strip width decreases as a power law approaching blowup

Numerical solutions support finite-time blowup in the model

Abstract

Motivated by the work on stagnation-point type exact solutions (with infinite energy) of 3D Euler fluid equations by Gibbon et al. (1999) and the subsequent demonstration of finite-time blowup by Constantin (2006) we introduce a one-parameter family of models of the 3D Euler fluid equations on a 2D symmetry plane. Our models are seen as a deformation of the 3D Euler equations which respects the variational structure of the original equations so that explicit solutions can be found for the supremum norms of the basic fields. The value of the model's parameter determines if there is finite-time blowup, and the singularity time can be computed explicitly in terms of the initial conditions and the model's parameter. We use a representative parameter value, for which the solution blows up in finite-time, as a benchmark for the systematic study of errors in numerical solutions. We compare numerical integrations of our "original" model with a "mapped" version of these equations. The mapped version is a globally regular (in time) system of equations, obtained via a bijective nonlinear mapping of time and fields from the original model equations (Bustamante 2011). We show that the mapped system's numerical solution increases accuracy in estimates of supremum norms and singularity time while entailing only a small additional computational cost. We study the Fourier spectrum of the model's numerical solution and find that the analyticity-strip width (a measure of the solution's analyticity) tends to zero as a power law in a finite time. This is in agreement with the finite-time blowup of the supremum norms, in the light of rigorous bounds stemming from the bridge (Bustamante & Brachet 2012) between the analyticity-strip method and the Beale-Kato-Majda type of theorems. We conclude by discussing the implications of this research on the analysis of numerical solutions of the 3D Euler fluid equations.