Traveling waves near Couette flow for the 2D Euler equation

TL;DR

This paper demonstrates the existence of a broad family of smooth traveling waves near Couette flow for the 2D Euler equation in low regularity spaces, contrasting with high-regularity results where such waves do not exist.

Contribution

It establishes the existence of nontrivial traveling waves in $H^s$ spaces with $s<3/2$, highlighting a new phenomenon in low regularity regimes.

Findings

Existence of large families of traveling waves near Couette flow.

Contrast with high-regularity spaces where such waves do not exist.

Implication for understanding stability and dynamics of 2D Euler flows.

Abstract

In this paper we reveal the existence of a large family of new, nontrivial and smooth traveling waves for the 2D Euler equation at an arbitrarily small distance from the Couette flow in $H^s$, with $s<3/2$, at the level of the vorticity. The speed of these waves is of order 1 with respect to this distance. This result strongly contrasts with the setting of very high regularity in Gevrey spaces (see arXiv:1306.5028), where the problem exhibits an inviscid damping mechanism that leads to relaxation of perturbations back to nearby shear flows. It also complements the fact that there not exist nontrivial traveling waves in the $H^{\frac{3}{2}+}$ neighborhoods of Couette flow (see arXiv:1004.5149).