Bounds on a singular attractor in Euler using vorticity moments

TL;DR

This paper introduces a new rescaling method for vorticity moments in 3D Euler equations to analyze vortex evolution, revealing bounds and growth behaviors that suggest potential singularity formation.

Contribution

It develops a novel rescaling approach for vorticity moments in Euler flows, linking moment growth to potential singularities and extending Navier-Stokes analysis techniques.

Findings

All rescaled moments grow over time.

Lower-order moments bound higher-order moments from above.

Growth of moments converges to singular bounds, indicating possible finite-time singularity.

Abstract

A new rescaling of the vorticity moments and their growth terms is used to characterise the evolution of anti-parallel vortices governed by the 3D Euler equations. To suppress unphysical instabilities, the initial condition uses a balanced profile for the initial magnitude of vorticity along with a new algorithm for the initial vorticity direction. The new analysis uses a new adaptation to the Euler equations of a rescaling of the vorticity moments developed for Navier-Stokes analysis. All rescaled moments grow in time, with the lower-order moments bounding the higher-order moments from above, consistent with new results from several Navier-Stokes calculations.Furthermore, if, as an inviscid flow evolves, this ordering is assumed to hold, then a singular upper bound on the growth of these moments can be used to provide a prediction of power law growth to compare against. There is a significant period where the growth of the highest moments converges to these singular bounds, demonstrating a tie between the strongest nonlinear growth and how the rescaled vorticity moments are ordered. The logarithmic growth of all the moments are calculated directly and the estimated singular times for the different $D_m$ converge to a common value for the simulation in the best domain.