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

초록

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 ∫0t∗(t∗-t)‖ω(t)‖L∞(B(x∗,R0))dt<+∞ for a ball B(x∗, R) 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) 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. © 2019, Springer-Verlag GmbH Germany, part of Springer Nature.

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