## Nonlinear stability of planar steady Euler flows associated with semistable solutions of elliptic problems

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- Trans. Amer. Math. Soc.
**375**(2022), 5071-5095 Request permission

## Abstract:

This paper is devoted to the study of nonlinear stability of steady incompressible Euler flows in two dimensions. We prove that a steady Euler flow is nonlinearly stable in $L^p$ norm of the vorticity if its stream function is a semistable solution of some semilinear elliptic problem with nondecreasing nonlinearity. The idea of the proof is to show that such a flow has strict local maximum energy among flows whose vorticities are rearrangements of a given function, with the help of an improved version of Wolansky and Ghil’s stability theorem. The result can be regarded as an extension of Arnol’d’s second stability theorem.## References

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## Additional Information

**Guodong Wang**- Affiliation: Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin 150001, People’s Republic of China
- Email: wangguodong@hit.edu.cn
- Received by editor(s): July 30, 2021
- Received by editor(s) in revised form: October 15, 2021, and December 31, 2021
- Published electronically: April 26, 2022
- Additional Notes: The author was supported by National Natural Science Foundation of China (12001135, 12071098) and China Postdoctoral Science Foundation (2019M661261, 2021T140163).
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**375**(2022), 5071-5095 - MSC (2020): Primary 35Q35, 76E30, 76B47
- DOI: https://doi.org/10.1090/tran/8652
- MathSciNet review: 4439499