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 (Ann PDE 5(4):1–192, 2019).
Original language | English (US) |
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Pages (from-to) | 61-137 |
Number of pages | 77 |
Journal | Archive For Rational Mechanics And Analysis |
Volume | 246 |
Issue number | 1 |
DOIs | |
State | Published - Oct 2022 |
Bibliographical note
Funding Information:Alexandru D. Ionescu was supported in part by NSF Grant DMS-2007008. Hao Jia was supported in part by NSF Grant DMS-1945179.
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH, DE, part of Springer Nature.