Abstract
In this note, we show that Luo-Hou’s ansatz for the self-similar solution to the axisymmetric solution to the 3D Euler equations leads to triviality of the solution under suitable decay condition of the blow-up profile. The equations for the blow-up profile reduces to an over-determined system of partial differential equations, whose only solution with decay is the trivial solution. We also propose a generalization of Luo-Hou’s ansatz. Using the vanishing of the normal velocity at the boundary, we show that this generalized self-similar ansatz also leads to a trivial solution. These results show that the self-similar ansatz may be valid either only in a time-dependent region which shrinks to the boundary circle at the self-similar rate, or under different boundary conditions at spatial infinity of the self-similar profile.
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References
Luo, G., Hou, T.: Potentially singular solutions of the 3D incompressible Euler equations, arXiv:1310.0497v2 (2013)
Majda, A., Bertozzi, A.: Vorticity and Incompressible Flow. Cambridge University Press, Cambridge (2002)
Titi, E.: lecture in NCTS, Hsinchu, 2012.12.13, arXiv:1401.1534.
Acknowledgments
This research was initiated when the authors visited Tsinghua Sanya International Mathematics Forum (TSIMF) in December 2013. We thank the referees for very helpful comments. Chae’s research is supported partially by NRF Grants Nos. 2006-0093854 and 2009-0083521. Tsai’s research is supported partially by NSERC grant 261356-13.
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Communicated by Peter Constantin.
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Chae, D., Tsai, TP. Remark on Luo-Hou’s Ansatz for a Self-similar Solution to the 3D Euler Equations. J Nonlinear Sci 25, 193–202 (2015). https://doi.org/10.1007/s00332-014-9225-6
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DOI: https://doi.org/10.1007/s00332-014-9225-6