Global existence of weak solutions to the three-dimensional Euler equations with helical symmetry

In this paper, we mainly investigate the weak solutions of the three-dimensional incompressible Euler equations with helical symmetry in the whole space when the helical swirl vanishes. Specifically, we establish the global existence of weak solutions when the initial vorticity lies in L1∩Lp with p>1. Our result extends the previous work [2], where the initial vorticity is compactly supported and belongs to Lp with p>4/3. The key ingredient in this paper involves the explicit analysis of Biot-Savart law with helical symmetry in domain R2×[−π,π] via the theories of singular integral operators and second order elliptic equations.