Long Time Dynamics for Helical Vortex Filament in Euler Flows

Abstract. We study the long time dynamics of a single helical vortex filament for the three-dimensional incompressible Euler equations under helical symmetry in the absence of vorticity stretching. The initial vorticity is assumed to be concentrated in an [Formula: see text]-neighborhood of a helix in the whole space [Formula: see text]. For any fixed time [Formula: see text] independent of [Formula: see text], the corresponding weak solution is shown to remain concentrated near a rotating helical filament throughout the interval [Formula: see text]. We further derive quantitative estimates for the filament motion and show that its angular velocity satisfies [Formula: see text]. The proof relies on asymptotic expansions for the helical Biot–Savart kernel in [Formula: see text] and on weighted energy–momentum functionals adapted to the helical symmetry, which allow us to control far-field effects and to propagate vorticity concentration on a fixed time interval.