2023/07/23 by Elgindi, Tarek M. · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn)
paper · doi:10.48550/arxiv.2307.12290
It is well-known that the first energy shell, S1c0:=\αcos(x+μ)+βcos(y+λ): α2+β2=c0 & (μ,λ)∈ℝ2\ of solutions to the 2d Euler equation is Lyapunov stable on \mathbbT2. This is simply a consequence of the conservation of energy and enstrophy. Using the idea of Wirosoetisno and Shepherd \citeWS, which is to take advantage of conservation of a properly chosen Casimir, we give a simple and quantitative proof of the L2 stability of single modes up to translation. In other words, each S1α,β:=\αcos(x+μ)+βcos(y+λ): (μ,λ)∈ℝ2\ is Lyapunov stable. Interestingly, our estimates indicate that the extremal cases α=0, β=0, and α=±β may be markedly less stable than the others.