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)=fu-q|lnε|inΩ,u=0on∂Ω.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} -\varepsilon ^2\text {div}(K_H(x)\nabla u)=f\left( u-q|\ln \varepsilon |\right) \ \text {in}\ \Omega ,\ \ \ u=0\ \text {on}\ \partial \Omega . \end{aligned}$$\end{document}By using the variational method, we show that for any 0<ε<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ 0<\varepsilon <1 $$\end{document}, there exist ground states concentrating near minimum points of q2det(KH)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ q^2\sqrt{det(K_H)} $$\end{document} as the parameter ε→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \varepsilon \rightarrow 0 $$\end{document}. These results show a striking difference with the 2D and the 3D axisymmetric Euler equation cases.