Linear Vortex Symmetrization: The Spectral Density Function

We investigate solutions of the 2d incompressible Euler equations, linearized around steady states which are radially decreasing vortices. Our main goal is to understand smoothness what we call spectral density function associated with operator, hope will be a step towards proving full nonlinear asymptotic stability The motivation for considering that it not possible describe vorticity or stream in terms one modulated profile. There fact two profiles, both at level physical and function. allows us identify these its leads pointwise decay consistent estimates first proved Bedrossian-Coti Zelati-Vicol [5].