Desingularization of 3D steady Euler equation with helical symmetry

TL;DR

This paper constructs steady 3D Euler flows with helical symmetry that approximate vortex filaments, using a semilinear elliptic problem and stream-function method to analyze their existence and asymptotic behavior.

Contribution

It introduces a novel approach to desingularize 3D Euler solutions with helical symmetry by constructing solutions concentrating near vortex filaments.

Findings

Existence of steady Euler flows with helical symmetry

Solutions concentrate near a point as parameter approaches zero

Qualitative properties of the constructed solutions

Abstract

In this paper, we study desingularization of steady solutions of 3D incompressible Euler equation with helical symmetry in a general helical domain. We construct a family of steady Euler flows with helical symmetry, such that the associated vorticities tend asymptotically to a helical vortex filament. The solutions are obtained by solving a semilinear elliptic problem in divergence form with a parameter. By using the stream-function method, we show the existence and asymptotic behavior of ground state solutions concentrating near a single point as the parameter $ \varepsilon\to 0 $. Qualitative properties of those solutions are also discussed.