Stationary Structures near the Kolmogorov and Poiseuille Flows in the 2d Euler Equations

TL;DR

This paper investigates stationary solutions near Kolmogorov and Poiseuille flows in 2D Euler equations, revealing new states near Kolmogorov flow and contrasting behaviors with monotone shear flows, with implications for Navier-Stokes dissipation.

Contribution

It constructs a large family of non-trivial stationary states near Kolmogorov flow, showing they are not shear flows, unlike in monotone shear flows, and analyzes the implications for related Navier-Stokes problems.

Findings

Existence of new stationary states near Kolmogorov flow.

Stationary states near Poiseuille flow are only shear flows.

Enhanced dissipation rates in linearized Navier-Stokes near these flows.

Abstract

We study the behavior of solutions to the incompressible $2d$ Euler equations near two canonical shear flows with critical points, the Kolmogorov and Poiseuille flows, with consequences for the associated Navier-Stokes problems. We exhibit a large family of new, non-trivial stationary states of analytic regularity, that are arbitrarily close to the Kolmogorov flow on the square torus $\mathbb{T}^2$. This situation contrasts strongly with the setting of some monotone shear flows, such as the Couette flow: in both cases the linearized problem exhibits an "inviscid damping" mechanism that leads to relaxation of perturbations of the base flows back to nearby shear flows. While this effect persists nonlinearly for suitably small and regular perturbations of some monotone shear flows, for the Kolmogorov flow our result shows that this is not possible. Our construction of these stationary states builds on a degeneracy in the global structure of the Kolmogorov flow on $\mathbb{T}^2$. In this regard both the Kolmogorov flow on a rectangular torus and the Poiseuille flow in a channel are very different, and we show that the only stationary states near them must indeed be shears, even in relatively low regularity $H^3$ resp. $H^{5+}$. In addition, we show that this behavior is mirrored closely in the related Navier-Stokes settings: the linearized problems near the Poiseuille and Kolmogorov flows both exhibit an enhanced rate of dissipation. Previous work by us and others shows that this effect survives in the full, nonlinear problem near the Poiseuille flow and near the Kolmogorov flow on rectangular tori, provided that the perturbations lie below a certain threshold. However, we show here that the corresponding result cannot hold near the Kolmogorov flow on $\mathbb{T}^2$.