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

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.