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

We consider the three-dimensional incompressible Euler equation a t Q + U center dot v Q - Q center dot vU center dot v U = 0 Q(x, 0) = Q0(x) 0 (x) the whole space R3. 3 . Under the assumption that Qz z is helical and in the absence of vorticity stretching, we prove the global well-posedness of weak solutions in L1 1 L 1 (R3). 3 ). Moreover, the vortex transport 1 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. 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.