Remarks on the blow-up criterion of the three-dimensional Euler equations

In this paper we prove that the finite time blow-up of classical solutions of the three-dimensional homogeneous incompressible Euler equations are controlled by the Besov space, , norm of the two componentsof the vorticity. For the axisymmetric flows with swirl we deduce that the blow-up of solution is controlled by the same Besov space norm of the angular componentof the vorticity. For a proof of these results we use the Beale-Kato-Majda criterion, and the special structure of the vortex stretching term in the vorticity formulation of the Euler equations.