The essential spectrum of advective equations

The geometric optics stability method is extended to a general class of linear advective PDE’s with pseudodifferential bounded perturbation. We give a new short proof of Vishik’s formula for the essential spectral radius. We show that every point in the dynamical spectrum of the corresponding bicharacteristicamplitude system contributes a point into the essential spectrum of the PDE. Generic spectral pictures are obtained in Sobolev spaces of sufficiently large smoothness. Applications to instability are presented.