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We develop a new structure of the Green's function of a second-order ellipticoperator in divergence form in a 2D bounded domain. Based on this structure andthe theory of rearrangement of functions, we construct concentratedtraveling-rotating helical vortex patches to 3D incompressible Euler equationsin an infinite pipe. By solving an equation for vorticity \begin{equation*}w=\frac{1}{\varepsilon^2}f_\varepsilon\left(\mathcal{G}_{K_H}w-\frac{\alpha}{2}|x|^2|\ln\varepsilon|\right)\ \ \text{in}\ \Omega \end{equation*} for small $ \varepsilon>0 $ andconsidering a certain maximization problem for the vorticity, where $\mathcal{G}_{K_H} $ is the inverse of an elliptic operator $ \mathcal{L}_{K_H}$ in divergence form, we get the existence of a family of concentrated helicalvortex patches, which tend asymptotically to a singular helical vortex filamentevolved by the binormal curvature flow. We also get nonlinear orbital stabilityof the maximizers in the variational problem under $ L^p $ perturbation when $p\geq 2. $

Structure of Green's function of elliptic equations and helical vortex patches for 3D incompressible Euler equations