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

  • Dongho ChaeEmail author
  • Jörg Wolf
Article

Abstract

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 \( \int \nolimits _{0}^{t_*} (t_*-t) \Vert \omega (t)\Vert _{ L^\infty (B(x_{ *}, R_0))}\,{\hbox {d}}t <+\infty \) 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 \( \Vert \omega (t)\Vert _{L^\infty (T(x_*, R))}= O\left( \frac{1}{(t_*-t)^\gamma }\right) \) as \(t\rightarrow t_*\), where \(\gamma <2\), and the torus \(T(x_*, R)\) does not touch the axis.

Notes

Acknowledgements

Chae was partially supported by NRF Grants 2016R1A2B3011647, while Wolf was supported by NRF Grants 2017R1E1A1A01074536.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsChung-Ang UniversitySeoulRepublic of Korea

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