Desingularization of 3D steady Euler equation with helical symmetry

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.