Helical vortices with small cross-section for 3D incompressible Euler equation

TL;DR

This paper constructs traveling-rotating helical vortices with small cross-sections for the 3D incompressible Euler equations, demonstrating their asymptotic approach to singular vortex filaments and including solutions with polygonal symmetry.

Contribution

It introduces a novel construction of helical vortex solutions with small cross-sections for the Euler equations, including configurations with polygonal symmetry.

Findings

Vortices tend asymptotically to singular helical vortex filaments.

Construction of solutions concentrating near multiple helical filaments.

Solutions exhibit polygonal symmetry.

Abstract

In this article, we construct traveling-rotating helical vortices with small cross-section to the 3D incompressible Euler equations in an infinite pipe, which tend asymptotically to singular helical vortex filament evolved by the binormal curvature flow. The construction is based on studying a general semilinear elliptic problem in divergence form \begin{equation*} \begin{cases} -\varepsilon^2\text{div}(K(x)\nabla u)= (u-q|\ln\varepsilon|)^{p}_+,\ \ &x\in \Omega,\\ u=0,\ \ &x\in\partial \Omega, \end{cases} \end{equation*} for small values of $ \varepsilon. $ Helical vortex solutions concentrating near several helical filaments with polygonal symmetry are also constructed.