Singularity Structures of Linear Inviscid Damping in a Channel

TL;DR

This paper analyzes the singularity structures in the linear inviscid damping of 2D Euler equations within a finite channel, revealing boundary-induced singularities and their dependence on initial data and background flow.

Contribution

It introduces a recursive framework to characterize spectrum density singularities and demonstrates their impact on the smoothness of the stream function near boundaries.

Findings

Stream function is smooth away from the boundary.

Singularities are caused by boundary and interior spectrum interactions.

Initial data and background flow influence regularity.

Abstract

This paper studies singularity structures of the linear inviscid damping of two-dimensional Euler equations in a finite periodic channel. We introduce a recursive definition of singularity structures which characterize the singularities of the spectrum density function from different sources: the free part and the boundary part of the Green function. As an application, we demonstrate that the stream function exhibits smoothness away from the channel's boundary, yet it presents singularities in close proximity to the boundary. The singularities arise due to the interaction of boundary and interior singularities of the spectrum density function. We also show that the behavior of the initial data and background flow have an impact on the regularity of different components of the stream function.