Desingularization of 3D steady Euler equations with helical symmetry

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 helical Euler flows, 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 -ε2div(KH(x)∇u)=f(u-q|lnε|)inΩ,u=0on∂Ω. By using the variational method, we show that for any 0 < ε< 1 , there exist ground states concentrating near minimum points of q2det(KH) as the parameter ε→ 0 . These results show a striking difference with the 2D and the 3D axisymmetric Euler equation cases.