Global well-posedness of weak solutions to the incompressible Euler equations with helical symmetry in $\mathbb{R}^3$

TL;DR

This paper proves the global existence and uniqueness of weak solutions to the 3D incompressible Euler equations with helical symmetry and no swirl, expanding understanding of such flows in mathematical fluid dynamics.

Contribution

It establishes the global well-posedness of weak solutions for the Euler equations under helical symmetry without swirl, a significant extension in the theory of fluid flows.

Findings

Global well-posedness of weak solutions proved

Vortex transport formula established

Solutions exist in $L^1_1 igcap L^{ abla}_1$ spaces

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*} in the whole space $\mathbb{R}^3$. Under the assumption that the initial velocity is helical and without swirl, we prove the global well-posedness of weak solutions in $L^1_1 \bigcap L^{\infty}_1(\mathbb{R}^3)$. The vortex transport formula is also obtained in our article.