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 Ω+U \cdot \nabla Ω-Ω\cdot \nabla U=0 \\ &Ω(x,0)=Ω_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 $\left[0,\frac{T}{\log \frac{1}{\epsilon }}\right)$.

30 pages