Existence of axisymmetric weak solutions of the 3-D Euler equations for near-vortex-sheet initial data

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 $omega_0$, we assumed that $omega_0/r$ belongs to $L(log L (Bbb R^3))^{alpha}$ with $alpha >1/2$, 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.