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Inviscid damping near the Couette flow in a channel

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arxiv 1808.04026 v2 pith:DVNOTRBF submitted 2018-08-13 math.AP math-phmath.MP

classification math.APmath-phmath.MP
keywords flowchannelcouetteshearwillflowsinteriorprove
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abstract

We prove asymptotic stability of shear flows in a neighborhood of the Couette flow for the 2D Euler equations in the domain $\T\times[0,1]$. More precisely we prove that if we start with a small and smooth perturbation (in a suitable Gevrey space) of the Couette flow, then the velocity field converges strongly to a nearby shear flow. The vorticity, which is initially assumed to be supported in the interior of the channel, will remain supported in the interior of the channel, will be driven to higher frequencies by the linear flow, and will converge weakly to $0$ as $t\to\infty$, modulo the shear flows (zero mode in $x$).

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  1. Stability threshold of the 2D Couette flow in Sobolev spaces

    math.AP 2019-08 conditional novelty 8.0 of 10

    For 2D Navier-Stokes near Couette flow, H^σ vorticity perturbations of size ≤ ε Re^{-1/3} are globally stable with inviscid damping and enhanced dissipation.

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