Inviscid damping of monotone shear flows for 2D inhomogeneous Euler equation with non-constant density in a finite channel

TL;DR

This paper proves nonlinear inviscid damping for certain monotone shear flows with non-constant density in 2D inhomogeneous Euler equations, demonstrating stability and decay of perturbations in a finite channel.

Contribution

It establishes the nonlinear inviscid damping for inhomogeneous 2D Euler flows with non-constant density, extending previous results to a broader class of flows.

Findings

Proves nonlinear inviscid damping for monotone shear flows with variable density.

Demonstrates stability of flows under Gevrey class perturbations.

Shows decay of perturbations over time in a finite channel.

Abstract

We prove the nonlinear inviscid damping for a class of monotone shear flows with non-constant background density for the two-dimensional ideal inhomogeneous fluids in $\mathbb{T}\times [0,1]$ when the initial perturbation is in Gevrey-$\frac{1}{s}$ ($\frac{1}{2}<s<1$) class with compact support.