On concentrated vortices of 3D incompressible Euler equations under helical symmetry: with swirl

TL;DR

This paper constructs and analyzes concentrated helical vortex solutions with swirl for 3D incompressible Euler equations, revealing their asymptotic behavior and connection to vortex filaments via variational methods.

Contribution

It introduces a new variational approach to construct helical vortices with swirl without orthogonality assumptions, linking solutions to vortex filament evolution.

Findings

Existence of traveling-rotating helical vortices with swirl.

Asymptotic convergence of vorticity to vortex filaments.

Development of a non-autonomous elliptic equation framework.

Abstract

In this paper, we consider the existence of concentrated helical vortices of 3D incompressible Euler equations with swirl. First, without the assumption of the orthogonality condition, we derive a 2D vorticity-stream formulation of 3D incompressible Euler equations under helical symmetry. Then based on this system, we deduce a non-autonomous second order semilinear elliptic equations in divergence form, whose solutions correspond to traveling-rotating invariant helical vortices with non-zero helical swirl. Finally, by using Arnold's variational method, that is, finding maximizers of a properly defined energy functional over a certain function space and proving the asymptotic behavior of maximizers, we construct families of concentrated traveling-rotating helical vortices of 3D incompressible Euler equations with non-zero helical swirl in infinite cylinders. As parameter $ \varepsilon\to0 $, the associated vorticity fields tend asymptotically to a singular helical vortex filament evolved by the binormal curvature flow.