Space quasi-periodic steady Euler flows close to the inviscid Couette flow

TL;DR

This paper constructs small amplitude, space quasi-periodic steady solutions to the 2D Euler equations near the Couette flow, revealing complex flow patterns like Kelvin's cat eye structures.

Contribution

It introduces a Nash-Moser scheme to prove existence of quasi-periodic solutions bifurcating from shear flows, a novel approach in this context.

Findings

Existence of space quasi-periodic steady Euler flows near Couette flow.

Flow streamlines exhibit Kelvin's cat eye-like patterns.

Solutions are stable for most shear parameters.

Abstract

We prove the existence of steady \emph{space quasi-periodic} stream functions, solutions for the Euler equation in vorticity-stream function formulation in the two dimensional channel ${\mathbb R}\times [-1,1]$. These solutions bifurcate from a prescribed shear equilibrium near the Couette flow, whose profile induces finitely many modes of oscillations in the horizontal direction for the linearized problem. Using a Nash-Moser implicit function iterative scheme, near such equilibrium we construct small amplitude, space reversible stream functions slightly deforming the linear solutions and retaining the horizontal quasi-periodic structure. These solutions exist for most values of the parameters characterizing the shear equilibrium. As a by-product, the streamlines of the nonlinear flow exhibit Kelvin's cat eye-like trajectories arising from the finitely many stagnation lines of the shear equilibrium.