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

Abstract

We develop a new structure of the Green's function of a second-order elliptic operator in divergence form in a 2D bounded domain. Based on this structure and the theory of rearrangement of functions, we construct concentrated traveling-rotating helical vortex patches to 3D incompressible Euler equations in 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 $ and considering 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 helical vortex patches, which tend asymptotically to a singular helical vortex filament evolved by the binormal curvature flow. We also get nonlinear orbital stability of the maximizers in the variational problem under $ L^p $ perturbation when $ p\geq 2. $