Axisymmetric Euler-α Equations without Swirl: Existence, Uniqueness, and Radon Measure Valued Solutions

The global existence of weak solutions for the three-dimensional axisymmetric Euler-α (also known as Lagrangian-averaged Euler-α) equations, without swirl, is established, whenever the initial unfiltered velocity v0 satisfies ∇×v0 r is a finite Randon measure with compact support. Furthermore, the global existence and uniqueness, is also established in this case provided ∇×v0 r ∈ L c (R) with p > 3 2 . It is worth mention that no such results are known to be available, so far, for the threedimensional Euler equations of ideal incompressible flows.