Operator algebras and the Fredholm spectrum of advective equations of linear hydrodynamics

In this paper we give a complete description of the Fredholm spectrum of the linearized 3D Euler equation in terms of the dynamical spectrum of the cocycle governing the evolution of shortwave perturbations. The argument is based on the pseudo differential representation of the Euler group and an application of an abstract isomorphism theorem for operator C∗-algebras. We further show that for all but countably many times the Fredholm spectrum of the group is rotationally invariant, and thus may consist of one or two concentric annuli. The results are obtained for a variety of other equations of ideal hydrodynamics.