Remark on the Stability of Energy Maximizers for the 2D Euler equation on ${\mathbb{T}}^2$

Abstract

It is well-known that the first energy shell, \[\mathcal{S}_1^{c_0}:=\{α\cos(x+μ)+β\cos(y+λ): α^2+β^2=c_0\,\, \&\,\, (μ,λ)\in\mathbb{R}^2\}\] 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 \[\mathcal{S}_1^{α,β}:=\{α\cos(x+μ)+β\cos(y+λ): (μ,λ)\in\mathbb{R}^2\}\] is Lyapunov stable. Interestingly, our estimates indicate that the extremal cases $\alpha =0,$ $\beta =0$, and $\alpha =\pm \beta$ may be markedly less stable than the others.