Inviscid damping near the Couette flow in a channel
Alexandru Ionescu, Hao Jia
TL;DR
The paper proves inviscid damping for small Gevrey-regular perturbations of Couette flow in a channel for the 2D Euler equations. It introduces a nonlinear adapted coordinate system and a sophisticated time-dependent, imbalanced weight framework to control transport and resonance, while carefully addressing boundary effects through localization and elliptic analysis. A bootstrap argument closes by showing weighted energies for the vorticity, the transformed vorticity, the coordinate functions, and the stream function all remain small and decay, yielding convergence to a nearby shear and decay of the perturbation. The work extends nonlinear stability results to a bounded domain with no-penetration boundaries and sharp Gevrey regularity, highlighting the role of boundary separation and precise energy methods in inviscid damping. The techniques combine a detailed spectral-transport analysis with weighted energy methods and boundary-aware elliptic estimates, providing a robust framework for stability near shear flows in confined geometries.
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 $x$).