Abstract
Sinusoidal flows are important explicit stationary solutions of the incompressible Euler equation on a flat two-torus. In this paper, we prove that sinusoidal flows related to least eigenfunctions of the negative Laplacian are, up to phase translations, nonlinearly stable under \(L^p\) norm of the vorticity for any \(1<p<+\infty \), which improves a classical stability result by Arnold. The key point of the proof is to distinguish least eigenstates with different amplitudes by using isovortical property of the Euler equation.
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Acknowledgements
G. Wang was supported by National Natural Science Foundation of China (12001135) and China Postdoctoral Science Foundation (2019M661261, 2021T140163). B. Zuo was supported by National Natural Science Foundation of China (12101154).
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Communicated by Y. Li.
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