Nonlinear inviscid damping for 2-D inhomogeneous incompressible Euler equations

TL;DR

This paper proves the global stability and convergence of shear flows near Couette flow for 2-D inhomogeneous incompressible Euler equations, demonstrating nonlinear inviscid damping in a new setting.

Contribution

It establishes the first global well-posedness and stability result for 2-D inhomogeneous incompressible Euler equations near Couette flow.

Findings

Velocity converges to shear flow close to Couette flow.

Vorticity is driven to small scales and weakly converges.

Global well-posedness in Gevrey class 2 for initial data close to Couette flow.

Abstract

We prove the asymptotic stability of shear flows close to the Couette flow for the 2-D inhomogeneous incompressible Euler equations on $\mathbb{T}\times \mathbb{R}$. More precisely, if the initial velocity is close to the Couette flow and the initial density is close to a positive constant in the Gevrey class 2, then 2-D inhomogeneous incompressible Euler equations are globally well-posed and the velocity converges strongly to a shear flow close to the Couette flow, and the vorticity will be driven to small scales by a linear evolution and weakly converges as $t\to \infty$. To our knowledge, this is the first global well-posedness result for the 2-D inhomogeneous incompressible Euler equations.