AN EXTENSION OF ARNOLD 2ND STABILITY THEOREM FOR THE EULER EQUATIONS

A general tool to prove nonlinear stability for stationary solutions o f infinite-dimensional Hamiltonian systems is the energy-Casimir metho d, proposed by Arnol'd for the two-dimensional, incompressible Euler e quations and generalized since to other flows. Arnol'd's stability the orems are based on a uniform estimate for the second variation of the energy-Casimir functional, leading to a strict convexity condition. In this paper we develop the method of supporting functionals and show t hat a local convexity condition, equivalent to the convexity of the qu asi-energy invariant of the linearized Euler equations, is also a suff icient condition for genuine stability of the fully nonlinear Euler eq uations.