Essential Spectrum of the Linearized Euler Equations and Evolution Semigroups

 

Abstract

We study spectral properties of the linearized Euler operator for an ideal incompressible fluid in dimensions two and three. The main tool in our analysis is the techniques of constructing approximate eigenfunctions that were recently developed for so-called evolution semigroups. The evolution semigroup is a semigroup of transfer-type operators induced by a given skew-product flow. We give information about the essential spectrum of the linearized Euler operator and describe the linearized hydrodynamic stability in terms of the spectrum of the linearized operator thus proving a spectral mapping theorem for the corresponding group. In particular, the boundaries of the essential spectrum are described in terms of Lyapunov-Oseledets exponents given by the Multiplicative Ergodic Theorem. Also, we relate the spectrum of the linearized Euler operator and Lyapunov exponents of the bicharacteristic amplitude system, a system of ordinary differential equations whose asymptotic behavior is responsible for the description of the essential spectrum of the linearized Euler operator.