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- W4385291281 abstract "It is well-known that the first energy shell, [mathcal{S}_1^{c_0}:={alpha cos(x+mu)+betacos(y+lambda): alpha^2+beta^2=c_0,, &,, (mu,lambda)inmathbb{R}^2}] of solutions to the 2d Euler equation is Lyapunov stable on $mathbb{T}^2$. This is simply a consequence of the conservation of energy and enstrophy. Using the idea of Wirosoetisno and Shepherd cite{WS}, which is to take advantage of conservation of a properly chosen Casimir, we give a simple and quantitative proof of the $L^2$ stability of single modes up to translation. In other words, each [mathcal{S}_1^{alpha,beta}:={alpha cos(x+mu)+betacos(y+lambda): (mu,lambda)inmathbb{R}^2}] is Lyapunov stable. Interestingly, our estimates indicate that the extremal cases $alpha=0,$ $beta=0$, and $alpha=pmbeta$ may be markedly less stable than the others." @default.
- W4385291281 created "2023-07-27" @default.
- W4385291281 creator A5028453038 @default.
- W4385291281 date "2023-07-23" @default.
- W4385291281 modified "2023-09-25" @default.
- W4385291281 title "Remark on the Stability of Energy Maximizers for the 2D Euler equation on $mathbb{T}^2$" @default.
- W4385291281 doi "https://doi.org/10.48550/arxiv.2307.12290" @default.
- W4385291281 hasPublicationYear "2023" @default.
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