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 $$\begin{aligned} w=\frac{1}{\varepsilon ^2}f_\varepsilon \left( \mathcal {G}_{K_H}w-\frac{\alpha }{2}|x|^2|\ln \varepsilon |\right) \ \ \text {in}\ \Omega \end{aligned}$$ 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 $$ p\ge 2. $$