REMARK ON THE STABILITY OF ENERGY MAXIMIZERS FOR THE 2D EULER EQUATION ON T2

It is well-known that the first energy shell, S-1(0)c := {alpha cos(x + mu) + beta cos(y + lambda) : alpha(2) + beta(2) = c(0) & (mu, lambda) is an element of R-2} of solutions to the 2d Euler equation is Lyapunov stable on T-2. This is simply a consequence of the conservation of energy and enstrophy. Using the idea of Wirosoetisno and Shepherd [17], 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 S-1(alpha,beta) := {alpha cos(x + mu) + beta cos(y + lambda) : (mu, lambda) is an element of R-2} is Lyapunov stable. Interestingly, our estimates indicate that the extremal cases alpha = 0, beta = 0, and alpha = +/-beta may be markedly less stable than the others.