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.
Categories: math.AP