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. [ABSTRACT FROM AUTHOR]