Abstract
In this article we show that the Euler equations, when linearized around a low frequency perturbation to Couette flow, exhibit norm inflation in Gevrey-type spaces as time tends to infinity. Thus, echo chains are shown to be a (secondary) linear instability mechanism. Furthermore, we develop a more precise analysis of cancellations in the resonance mechanism, which yields a modified exponent in the high frequency regime. This allows us, in addition, to remove a logarithmic constraint on the perturbations present in prior works by Bedrossian, Deng and Masmoudi, and to construct solutions which are initially in a Gevrey class for which the velocity asymptotically converges in Sobolev regularity but diverges in Gevrey regularity.
Highlights
In this article our aim is to develop a further understanding of the long-time asymptotic behavior of the 2D Euler equations near Couette flow∂t ω + y∂x ω + v · ∇ω = 0 (1)in an infinite channel T × R
Our main result is that this model exhibits the same echo chains as the full nonlinear Euler equations near Couette flow, but further exhibits modified scattering and linear inviscid damping in the sense the velocity perturbation strongly converges as t → ∞, but that the vorticity diverges in H s for any s > −1
– Any instability will have to sustain a sequence of infinitely many separate echo chains for a sequence of times tending to infinity to ensure that the flow is not asymptotically stable after all
Summary
Existing works study the infinite time asymptotic stability of the vorticity in higher Sobolev regularity or Gevrey regularity, from which (linear) inviscid damping follows as a corollary This is a strictly stronger condition than the physically observed phenomenon of inviscid damping, which is the convergence of the velocity field. Our main result is that this model exhibits the same echo chains as the full nonlinear Euler equations near Couette flow (in Gevrey regularity, see [2,5]), but further exhibits modified scattering and linear inviscid damping in the sense the velocity perturbation strongly converges as t → ∞, but that the vorticity diverges in H s for any s > −1. A key effort of this article in Sections 4 and 5 is to remove this constraint and show that solutions behave qualitatively differently in that high frequency regime than in the previously considered low frequency case
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