Long time dynamics for helical vortex filament in Euler flows

Abstract

We consider the three-dimensional incompressible Euler equation \begin{equation*}\left\{\begin{aligned} &\partial_t \Omega+U \cdot \nabla \Omega-\Omega\cdot \nabla U=0 \\ &\Omega(x,0)=\Omega_0(x) \end{aligned}\right. \end{equation*} under the assumption that $\Omega^z$ is helical and in the absence of vorticity stretching. Assuming that the initial vorticity $\Omega_0$ is primarily concentrated within an $\epsilon$ neighborhood of a helix $\Gamma_0$, we prove that its solution $\Omega(\cdot,t)$ remain concentrated near a helix $\Gamma(t)$ for any $t \in [0,T)$, where $\Gamma(t)$ can be interpreted as $\Gamma_0$ rotating around the $x_3$ axis with a speed $V=C\log \frac{1}{\epsilon}+O(1)$. It should be emphasized that the dynamics for the helical vortex filament are exhibited on the time interval $[0,T)$, which is longer than $\left[0, \frac{T}{\log\frac{1}{\epsilon}}\right)$.