Global existence of a weak solution of the incompressible Euler equations with helical symmetry and $L^p$ vorticity

TL;DR

This paper proves the global existence of weak solutions to the 3D Euler equations with helical symmetry and $L^p$ vorticity, extending previous results to unbounded domains and less restrictive initial vorticity conditions.

Contribution

It extends the existence theory of helical Euler flows to initial vorticities in $L^p$, for $p > 4/3$, in full space, without swirl.

Findings

Established global weak solutions for helical symmetry with $L^p$ vorticity.

Extended prior work from bounded helical pipes to full space.

Demonstrated solutions exist for initial vorticity in $L^p$, $p > 4/3$.

Abstract

We prove the global existence of a helical weak solution of the 3D Euler equations, in full space, for an initial velocity with helical symmetry, without swirl and whose initial vorticity is compactly supported in the axial plane and belongs to $L^p$, for some $p>\frac{4}{3}$. This result is an extension of the existence part of the work of B. Ettinger and E. Titi (SIAM J. Math Anal. 41(2009) 269-296), who studied well-posedness of the Euler equations with helical symmetry without swirl, with bounded initial vorticity, in a helical pipe.