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

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.