THE SPECTRUM OF THE 2ND VARIATION OF THE ENERGY FOR AN IDEAL INCOMPRESSIBLE FLUID

The Arnold theory of hydrodynamic stability suggests the study of the second variation of the energy on the submanifold of isovorticial vect or fields. We show that, in three or more dimensions, the spectrum of the associated Hermitian operator generally ranges from -infinity to i nfinity. The sole exceptional case is where the velocity field is harm onic (solenoidal and irrotational) in which case the Hessian is identi cally zero.