Stationary Structures Near the Kolmogorov and Poiseuille Flows in the 2d Euler Equations
Abstract

We study the behavior of solutions to the incompressible 2dEuler 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$$\mathbb {T}^2$$${T}^{2}$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$$\mathbb {T}^2$$${T}^{2}$. Our construction of the new stationary states builds on a degeneracy in the global structure of the Kolmogorov flow on$$\mathbb {T}^2$$${T}^{2}$, 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 more »

Authors:
; ;
Publication Date:
NSF-PAR ID:
10394858
Journal Name:
Archive for Rational Mechanics and Analysis
Volume:
247
Issue:
1
ISSN:
0003-9527
Publisher:
National Science Foundation
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2. Abstract

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5. Abstract

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