Title: Global existence of a weak solution of the incompressible Euler equations with helical symmetry and $L^p$ vorticity
Authors: Anne Bronzi, Helena Judith Nussenzveig Lopes and Milton da Costa Lopes Filho
Issue: Volume 64 (2015), Issue 1, 309-341
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 [B.\ Ettinger and E.\:S.\ Titi, \textit{Global existence and uniqueness of weak solutions of three-dimensional Euler equations with helical symmetry in the absence of vorticity stretching}, SIAM J. Math. Anal. \textbf{41} (2009), no. 1, 269--296], who studied well-posedness of the Euler equations with helical symmetry without swirl, with bounded initial vorticity, in a helical pipe.](https://iumj.s3-us-west-2.amazonaws.com/abstracts/5467.png)