Linear inviscid damping for monotone shear flows in a finite periodic channel, boundary effects, blow-up and critical Sobolev regularity

TL;DR

This paper analyzes the behavior of monotone shear flows in a finite periodic channel, revealing how boundary effects cause singularities and determining stability and blow-up in different Sobolev spaces, impacting nonlinear damping understanding.

Contribution

It provides a detailed description of singularity formation and establishes stability and blow-up results in fractional Sobolev spaces for shear flows in finite channels.

Findings

Boundary effects lead to derivative singularities.

Stability in sub-critical Sobolev spaces.

Blow-up in super-critical Sobolev spaces.

Abstract

In a previous article, \cite{Zill3}, we have established linear inviscid damping for a large class of monotone shear flows in a finite periodic channel and have further shown that boundary effects asymptotically lead to the formation of singularities of derivatives of the solution. As the main results of this article, we provide a detailed description of the singularity formation and establish stability in all sub-critical fractional Sobolev spaces and blow-up in all super-critical spaces. Furthermore, we discuss the implications of the blow-up to the problem of nonlinear inviscid damping in a finite periodic channel, where high regularity would be essential to control nonlinear effects.