On Recent Developments in the Spectral Problem for the Linearized Euler Equation

The purpose of this article is to survey results concerning the unstable spectrum of the Euler equation linearized about a steady state. The Euler equations of the motion of an inviscid, incompressible fluid are the basic equations of fluid mechanics and they have been the object of much study by mathematicians over the centuries since Euler ”unveiled” them in 1755. However, many significant problems concerning the Euler equation remain unsolved. Some of these problems arise from the particularly challenging nature of the nonlinearity of the equations. But even the linearized equations give rise to complex issues and open questions. Among the most fundamental of these is the structure of the spectrum of the linear Euler operator. Because this operator is, generally, degenerate, non-self adjoint and non-elliptic the standard well known theorems concerning the spectra of elliptic operators do not apply. It has been necessary to develop specialized tools using several branches of mathematics to investigate the spectrum of the Euler operator. Techniques used include those from PDE, functional analysis, operator theory and ODE. In this paper we will discuss our understanding of results achieved and the problems that remain open. A prime motivation for the study of the spectrum of the Euler equation linearized about a steady flow u(x) is the question of (linear) stability or instability of this flow. The ”classical” works on the subject in the 19 and earlier 20 centuries concerned an examination of the discrete unstable spectrum (i.e. eigenvalues for a linear PDE with positive real part) for a very special class of flows u(x) . We discuss