Linear vortex symmetrization: the spectral density function

Abstract

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

arXiv comment64 pages

References in corpus (6)

• Global stability of vortex solutions of the two-dimensional Navier-Stokes equation
• Inviscid damping near the Couette flow in a channel
• Large time behavior and asymptotic stability of the two-dimensional Euler and linearized Euler equations
• Linear inviscid damping for monotone shear flows in a finite periodic channel, boundary effects, blow-up and critical Sobolev regularity
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Cited by in corpus (2)

• A dynamical approach to the study of instability near Couette flow
• Uniform linear inviscid damping and enhanced dissipation near monotonic shear flows in high Reynolds number regime (I): the whole space case