Existence of Axisymmetric Weak Solutions of the 3-D Euler Equations for Near-Vortex-Sheet Initial Data
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We study the initial value problem for the 3-D Euler equation when the fluid is inviscid and incompressible, and flows with axisymmetry and without swirl. On the initial vorticity ω0, we assumed that ω0/r belongs to L(log L(R3))ɑ with ɑ > 1/2, where r is the distance to an axis of symmetry. To prove the existence of weak global solutions, we prove first a new a priori estimate for the solution.