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

Christian Zillinger

TL;DR

This work analyzes linear inviscid damping for monotone shear flows in a finite periodic channel, highlighting how boundary effects generate derivative singularities that dictate sharp Sobolev-regularity thresholds. By formulating the problem in a flow-following coordinate system and employing a scattering framework plus fractional Sobolev tools, the authors establish subcritical stability and supercritical blow-up with explicit exponents: s=3/2 for general perturbations and s=5/2 for perturbations with vanishing Dirichlet data, accompanied by logarithmic boundary singularities. They develop boundary-correction and elliptic-regularity mechanisms, study boundary layers, and discuss the implications for nonlinear inviscid damping, showing consistency in some regimes but fundamental obstructions to high-regularity damping in finite channels. The results illuminate the essential role of boundary behavior in determining damping rates, scattering, and nonlinear prospects, with direct impact on how such problems are approached in finite domains.

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.

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

TL;DR

This work analyzes linear inviscid damping for monotone shear flows in a finite periodic channel, highlighting how boundary effects generate derivative singularities that dictate sharp Sobolev-regularity thresholds. By formulating the problem in a flow-following coordinate system and employing a scattering framework plus fractional Sobolev tools, the authors establish subcritical stability and supercritical blow-up with explicit exponents: s=3/2 for general perturbations and s=5/2 for perturbations with vanishing Dirichlet data, accompanied by logarithmic boundary singularities. They develop boundary-correction and elliptic-regularity mechanisms, study boundary layers, and discuss the implications for nonlinear inviscid damping, showing consistency in some regimes but fundamental obstructions to high-regularity damping in finite channels. The results illuminate the essential role of boundary behavior in determining damping rates, scattering, and nonlinear prospects, with direct impact on how such problems are approached in finite domains.

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.

Paper Structure

This paper contains 19 sections, 48 theorems, 398 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Results for general perturbations
  4. Results for perturbations with vanishing Dirichlet data
  5. Consistency and implications for the nonlinear problem
  6. Outline of the article
  7. Preliminaries
  8. The linearized 2D Euler equations in scattering formulation
  9. A constant coefficient model for the damping mechanism
  10. Fractional Sobolev spaces
  11. Boundary effects and sharp stability results
  12. Stability in $H^{3/2-}$ and boundary perturbations
  13. Boundary corrections
  14. Elliptic control
  15. Stability in $H^{5/2-}$
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $U$ be such that $\frac{1}{U'}, U" \in W^{2,\infty}(\mathbb{T}_{L}\times[a,b])$, let $W$ be the solution of eq:LEusualcoord and let $v=\nabla^{\bot}\phi$ be the associated velocity field. Then the following statements hold:

Theorems & Definitions (86)

  • Theorem 1.1: Damping using regularity,Zill3
  • Theorem 1.2: Stability in $H^{1}$ and $H^{2}$, Zill3
  • Corollary 1.1
  • Theorem 1.3
  • Corollary 1.2
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1: Linearized 2D Euler equations in scattering formulation
  • Theorem 2.1: Zill3
  • Proposition 2.1: Fractional Sobolev space on $\mathbb{R}$; hitch
  • ...and 76 more