Traveling waves near shear flows for the inhomogeneous Euler equations with non-constant density

TL;DR

This paper studies the existence of traveling wave solutions near shear flows with variable density in 2D inhomogeneous Euler equations, revealing conditions for their existence and nonexistence based on regularity and spectral properties.

Contribution

It constructs traveling wave solutions at certain regularities and proves nonexistence in higher regularity spaces under spectral conditions, advancing understanding of wave behavior in variable density flows.

Findings

Traveling waves exist at specific regularities ($H^{5/2- au}$ and $H^{3/2- au}$).

Inviscid damping fails at these regularities.

Traveling waves do not exist at higher regularities when the Rayleigh operator has no eigenvalues.

Abstract

We investigate the existence and nonexistence of traveling wave solutions near monotonic shear flows with non-constant background density for the two-dimensional inhomogeneous Euler equations in a finite channel. For any small $\tau>0$, first, we construct nontrivial traveling waves with velocity and density in $H^{5/2-\tau}$ and $H^{3/2-\tau}$, respectively, showing that inviscid damping fails at these regularities. Second, when the distorted Rayleigh operator has no eigenvalues, we prove that such traveling wave solutions cannot exist in higher regularity spaces ($H^{5/2+\tau}$ for velocity and $H^{3/2+\tau}$ for density).