Global well-posedness of weak solutions to the incompressible Euler equations with helical symmetry in R3

Abstract

We consider the three-dimensional incompressible Euler equation$\{\begin{array}{cc}& {\partial }_t\Omega +U\cdot \nabla \Omega -\Omega \cdot \nabla U=0\\ & \Omega (x,0)={\Omega }_0\left(x\right)\end{array}$ in the whole space $R^3$. Under the assumption that ${\Omega }^z$ is helical and in the absence of vorticity stretching, we prove the global well-posedness of weak solutions in $L_1^1\bigcap L_1^{\infty }\left(R^3\right)$. Moreover, the vortex transport formula and the conservation of the energy and the second momentum are also obtained in our article, which will serve as valuable tools in our subsequent exploration of the dynamics of helical vortex filaments.