Bursting Dynamics of the 3D Euler Equations in Cylindrical Domains

TL;DR

This paper investigates the bursting behavior of 3D Euler equations in cylindrical domains with large vorticity, revealing how resonances can lead to singularity-like events in fluid flow.

Contribution

It introduces a novel analysis of resonant interactions in cylindrical geometries, connecting fast oscillations to nonlinear depletion and burst phenomena in Euler flows.

Findings

Resonances deplete Euler nonlinearity.

Resonant systems exhibit homoclinic cycles.

Orbits near these cycles cause solution bursts.

Abstract

A class of three-dimensional initial data characterized by uniformly large vorticity is considered for the Euler equations of incompressible fluids. The fast singular oscillating limits of the Euler equations are studied for parametrically resonant cylinders. Resonances of fast swirling Beltrami waves deplete the Euler nonlinearity. The resonant Euler equations are systems of three-dimensional rigid body equations, coupled or not. Some cases of these resonant systems have homoclinic cycles, and orbits in the vicinity of these homoclinic cycles lead to bursts of the Euler solution measured in Sobolev norms of order higher than that corresponding to the enstrophy.