91ÌÒÉ«

In this talk, we first revisit the vortex filament conjecture for three-dimensional incompressible Euler flows with helical symmetry and no swirl. By adapting gluing methods, we construct the first smooth helical vortex filament in the whole space $\mathbb{R}^3$, whose cross-sectional vorticity remains compactly supported in $\mathbb{R}^2$ for all times. We then extend the construction to clusters of vortex helices, corresponding to collapsing multi-vortex helical configurations. Finally, we discuss the intriguing leapfrogging phenomenon for helical vortex filaments, in which several interacting vortex helices with a common symmetry axis alternately overtake one another while preserving their coherent structure. This is joint work with Monica Musso.