Stationary structures near the Kolmogorov and Poiseuille flows in the 2d Euler equations
File(s)
Published version
Author(s)
Coti Zelati, Michele
Elgindi, Tarek M
Widmayer, Klaus
Type
Journal Article
Abstract
We study the behavior of solutions to the incompressible 2d Euler equations near two canonical shear flows with critical points, the Kolmogorov and Poiseuille flows, with consequences for the associated Navier–Stokes problems. We exhibit a large family of new, non-trivial stationary states that are arbitrarily close to the Kolmogorov flow on the square torus T2 in analytic regularity. This situation contrasts strongly with the setting of some monotone shear flows, such as the Couette flow: there the linearized problem exhibits an “inviscid damping” mechanism that leads to relaxation of perturbations of the base flows back to nearby shear flows. Our results show that such a simple description of the long-time behavior is not possible for solutions near the Kolmogorov flow on T2. Our construction of the new stationary states builds on a degeneracy in the global structure of the Kolmogorov flow on T2, and we also show a lack of correspondence between the linearized description of the set of steady states and its true nonlinear structure. Both the Kolmogorov flow on a rectangular torus and the Poiseuille flow in a channel are very different. We show that the only stationary states near them must indeed be shears, even in relatively low regularity. In addition, we show that this behavior is mirrored closely in the related Navier–Stokes settings: the linearized problems near the Poiseuille and Kolmogorov flows both exhibit an enhanced rate of dissipation. Previous work by us and others shows that this effect survives in the full, nonlinear problem near the Poiseuille flow and near the Kolmogorov flow on rectangular tori, provided that the perturbations lie below a certain threshold. However, we show here that the corresponding result cannot hold near the Kolmogorov flow on T2.
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Date Issued
2023-02-01
Date Acceptance
2023-01-11
Citation
Archive for Rational Mechanics and Analysis, 2023, 247 (1)
ISSN
0003-9527
Publisher
Springer
Journal / Book Title
Archive for Rational Mechanics and Analysis
Volume
247
Issue
1
Copyright Statement
© The Author(s) (2023) Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
License URL
Identifier
http://arxiv.org/abs/2007.11547v1
Subjects
35Q31, 76E05
math.AP
math.AP
Publication Status
Published
Article Number
12
Date Publish Online
2023-02-01