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

TL;DR

This paper introduces a new structure for the Green's function of elliptic operators and constructs concentrated helical vortex patches for 3D Euler equations, demonstrating their stability and asymptotic behavior.

Contribution

It develops a novel Green's function structure and constructs stable, concentrated helical vortex patches for 3D Euler equations using variational methods.

Findings

Existence of concentrated helical vortex patches that tend to vortex filaments.

Construction of vortex patches via a new Green's function structure.

Proven nonlinear orbital stability of the vortex patches.

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. $