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) \| ω(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 $t=t_{\ast }$ in the torus $T(x_{\ast },R)$ generated by rotation of the ball $B(x_{\ast },R_0)$ around the axis. This implies that possible singularity at $t=t_{\ast }$ in the torus $T(x_{\ast },R)$ is excluded if the vorticity satisfies the blow-up rate $\Vert ø(t){\Vert }_{L^{\infty }(T(x_{\ast },R))}=O\left(\frac{1}{(t_{\ast }-t{)}^{\gamma }}\right)$ as $t\to t_{\ast }$, where $\gamma <2$ and the torus $T(x_{\ast },R)$ does not touch the axis.

47 pages

References in corpus (1)

• An incompressible 2D didactic model with singularity and explicit solutions of the 2D Boussinesq equations