Removing Type II Singularities Off the Axis for the Three Dimensional Axisymmetric Euler Equations

In this paper we obtain new local blow-up criterion for smooth axisymmetric solutions to the three dimensional incompressible Euler equation. If the vorticity satisfies ∫ 0 t ∗ ( t ∗ - t ) ‖ ω ( t ) ‖ L ∞ ( B ( x ∗ , R 0 ) ) d t < + ∞ for a ball B ( x ∗ , R 0 ) away from the axis of symmetry, then there exists no singularity at t = t ∗ in the torus T ( x ∗ , R ) generated by rotation of the ball B ( x ∗ , R 0 ) around the axis. This implies that possible singularity at t = t ∗ in the torus T ( x ∗ , R ) is excluded if the vorticity satisfies the blow-up rate ‖ ω ( t ) ‖ L ∞ ( T ( x ∗ , R ) ) = O 1 ( t ∗ - t ) γ as t → t ∗ , where γ < 2 , and the torus T ( x ∗ , R ) does not touch the axis.