On strong convergence to 3-D axisymmetric vortex sheets

We consider the 3-D axisymmetric incompressible Euler equations without swirls with vortex-sheets initial data. It is proved that the approximate solutions, generated by smoothing the initial data, converge strongly in provided that they have strong convergence in the region away from the symmetry axis. This implies that if there would appear singularity or energy lost in the process of limit for the approximate solutions, it then must happen in the region away from the symmetry axis. There is no restriction on the signs of initial vorticity here. In order to exclude the possible concentrations on the symmetry axis, we use the special structure of the equations for axisymmetric flows and careful choice of test functions.