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The global existence of weak solutions for the three-dimensional axisymmetricEuler-$\alpha$ (also known as Lagrangian-averaged Euler-$\alpha$) equations,without swirl, is established, whenever the initial unfiltered velocity $v_0$satisfies $\frac{\nabla \times v_0}{r}$ is a finite Randon measure with compactsupport. Furthermore, the global existence and uniqueness, is also establishedin this case provided $\frac{\nabla \times v_0}{r} \in L^p_c(\mathbb{R}^3)$with $p>{3/2}$. It is worth mention that no such results are known to beavailable, so far, for the three-dimensional Euler equations of idealincompressible flows.

Axisymmetric Euler-$\alpha$ Equations without Swirl: Existence, Uniqueness, and Radon Measure Valued Solutions