Linear Vortex Symmetrization: The Spectral Density Function

We investigate solutions of the 2 d 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 ste...

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Bibliographic Details
Published in:Archive for rational mechanics and analysis 2022-10, Vol.246 (1), p.61-137
Main Authors: Ionescu, Alexandru D., Jia, Hao
Format: Article
Language:English
Subjects:
Classical Mechanics
Complex Systems
Decay
Density
Digital Object Identifier
Euler-Lagrange equation
Fluid- and Aerodynamics
Linearization
Mathematical and Computational Physics
Physics
Physics and Astronomy
Reynolds number
Smoothness
Spectral density function
Theoretical
Vortices
Vorticity
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Summary:We investigate solutions of the 2 d 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 (Ann PDE 5(4):1–192, 2019).
ISSN:0003-9527
1432-0673
DOI:10.1007/s00205-022-01815-y