Removing Type II singularities off the axis for the 3D axisymmetric Euler equations

Abstract: We prove local blow-up criterion for smooth axisymmetric solutions to the 3D incompressible Euler equation. If the vorticity satisfies $ \intl_{0}^{t_*} (t_-t) | \omega (t)|{ L^\infty(B(x{ \ast}, R_0))} dt <+\infty$ for a ball $B(x_{ \ast}, R_0)$ 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 $B(x_{ \ast}, R_0)$ 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 $ |\o (t)|{L^\infty (T(x, R))}= O\left(\frac{1}{(t_-t)^\gamma}\right)$ as $t\to t_$, where $\gamma <2$ and the torus $T(x_*, R)$ does not touch the axis.