Nonlinear stability of plane ideal flows in a periodic channel
Guodong Wang
TL;DR
This work addresses the nonlinear stability of plane ideal flows for the 2D Euler equations in a finite periodic channel by developing a compactness-based variational framework that leverages rearrangement theory and the isovortical property, avoiding Arnold’s energy-Casimir definiteness. The authors establish two Burton-type stability theorems: the first yields stability under a relaxed spectral bound $0\le\min g'\le\max g'<\Lambda_1$, while the second provides orbital stability with a decomposition of the flow into a first eigenspace component and a vertical shear, under $\max g'\le\Lambda_1$. They prove the existence of stable non-shear flows when the channel aspect ratio satisfies $H/L\le\sqrt{3}/2$, and obtain rigidity results from the structure of maximizers of the kinetic energy within rearrangement classes. The results deepen the understanding of nonlinear stability in symmetric domains by exploiting the Euler invariants and rearrangement geometry, with potential implications for constructing and validating stable zonal and non-zonal flows in bounded geometries.
Abstract
In this paper, we establish two stability theorems for steady or traveling solutions of the two-dimensional incompressible Euler equation in a finite periodic channel, extending Arnold's classical work from the 1960s. Compared to Arnold's approach, we employ a compactness argument rather than relying on the negative definiteness of the energy-Casimir functional. The isovortical property of the Euler equation and Burton's rearrangement theory play an essential role in our analysis. As a corollary, we prove for the first time the existence of a class of stable non-shear flows when the ratio of the channel's height to its length is less than or equal to $\sqrt{3}/2.$ Two rigidity results are also obtained as byproducts.