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Long time dynamics for helical vortex filament in Euler flows

Dengjun Guo, Lifeng Zhao

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 $Ω^z$ is helical and in the absence of vorticity stretching. Assuming that the initial vorticity $Ω_0$ is primarily concentrated within an $ε$ neighborhood of a helix $Γ_0$, we prove that its solution $Ω(\cdot,t)$ remain concentrated near a helix $Γ(t)$ for any $t \in [0,T)$, where $Γ(t)$ can be interpreted as $Γ_0$ rotating around the $x_3$ axis with a speed $V=C\log \frac{1}ε+O(1)$. It should be emphasized that the dynamics for the helical vortex filament are exhibited on the time interval $[0,T)$, which is longer than $\left[0, \frac{T}{\log\frac{1}ε}\right)$.

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 is helical and in the absence of vorticity stretching. Assuming that the initial vorticity is primarily concentrated within an neighborhood of a helix , we prove that its solution remain concentrated near a helix for any , where can be interpreted as rotating around the axis with a speed . It should be emphasized that the dynamics for the helical vortex filament are exhibited on the time interval , which is longer than .
Paper Structure (12 sections, 21 theorems, 269 equations)

This paper contains 12 sections, 21 theorems, 269 equations.

Table of Contents

  1. Introduction
  2. Mathematical preliminaries and main result.
  3. Helical functions and vector fields.
  4. Function spaces and the definition of weak solutions
  5. Biot-Savart law
  6. Stream function
  7. Velocity field
  8. Physical quantities for helical Euler equation
  9. Long time dynamics for helical vortex filament
  10. Concentration of the vorticity
  11. Dynamics for $p_{\epsilon}^*(t)$
  12. Dynamics for $p_{e}(t)$

Key Result

Theorem 1.1

Given $T>0$, let $\,w_{\epsilon}(x,t)$ be a weak solution to the two-dimensional helical Euler equation eq 2euler with initial data $w_{\epsilon,0}(x)$. Assuming in addition that $w_{\epsilon,0}(x) \in L^1_4 \bigcap L^{\infty}_4$ satisfies Assumption as initial data, then for any $t \in [0,T)$ and $ holds whenever $\epsilon \le \epsilon_0$. Furthermore, define then for any $t\in [0,T)$, there hol

Theorems & Definitions (39)

  • Theorem 1.1
  • Remark 1.2
  • Definition 2.1: helical function
  • Definition 2.2: helical vector field
  • Lemma 2.3: GZ1
  • Proposition 2.4: GZ1 Two-dimensional helical Euler equation
  • Lemma 2.5
  • Definition 2.6
  • Definition 2.7
  • Theorem 2.8: GZ1
  • ...and 29 more