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 Ω+U \cdot \nabla Ω+Ω\cdot \nabla U=0 \\ &Ω(x,0)=Ω_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 $L_1^1\bigcap L_1^{\infty }({\mathbb{R}}^3)$. The vortex transport formula is also obtained in our article.

44 pages