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Linear inviscid damping near monotone shear flows

Hao Jia

TL;DR

The paper provides an elementary, self-contained proof of sharp linear inviscid damping for monotone shear flows in a periodic channel, and derives precise $L^{ty}$-level asymptotics for the linearized Euler problem. By introducing spectrally adapted function spaces that capture singularities of generalized eigenfunctions, the authors obtain optimal decay rates and a detailed decomposition of the flow into main oscillatory, boundary, and remainder terms. A limiting absorption principle and careful regularity analysis of the spectral measure underpin the results, including explicit boundary-term contributions that influence the asymptotics. The work clarifies the role of boundary effects in damping dynamics and lays groundwork for potential nonlinear extensions in regimes where vorticity remains away from the boundary or Gevrey regularity is available.

Abstract

We give an elementary proof of sharp decay rates and the linear inviscid damping near monotone shear flow in a periodic channel, first obtained in [14]. We shall also obtain the precise asymptotics of the solutions, measured in the space $L^{\infty}$.

Linear inviscid damping near monotone shear flows

TL;DR

The paper provides an elementary, self-contained proof of sharp linear inviscid damping for monotone shear flows in a periodic channel, and derives precise -level asymptotics for the linearized Euler problem. By introducing spectrally adapted function spaces that capture singularities of generalized eigenfunctions, the authors obtain optimal decay rates and a detailed decomposition of the flow into main oscillatory, boundary, and remainder terms. A limiting absorption principle and careful regularity analysis of the spectral measure underpin the results, including explicit boundary-term contributions that influence the asymptotics. The work clarifies the role of boundary effects in damping dynamics and lays groundwork for potential nonlinear extensions in regimes where vorticity remains away from the boundary or Gevrey regularity is available.

Abstract

We give an elementary proof of sharp decay rates and the linear inviscid damping near monotone shear flow in a periodic channel, first obtained in [14]. We shall also obtain the precise asymptotics of the solutions, measured in the space .

Paper Structure

This paper contains 11 sections, 13 theorems, 207 equations.

Table of Contents

  1. Introduction
  2. Equations and the main result
  3. Main idea of proof
  4. Boundedness of the operator $T_{k,y_0,\epsilon}$
  5. The limiting absorption principle
  6. Regularity of the spectral measure
  7. Proof of the main theorem
  8. Proof of Theorem \ref{['thm']}
  9. Two technical lemmas
  10. Proof of Lemma \ref{['P0']}
  11. Proof of Lemma \ref{['P18']}

Key Result

Theorem 1.1

Let $\omega$ be a smooth solution to main with associated velocity field $u=(u^x,u^y)$, stream function $\psi$, and initial data $\omega_0\in H^3(\mathbb{T}\times[0,1])$. Assume that $\int_{\mathbb{T}}\omega_0(x,y)\,dx=0$ and that $\omega_0$ belongs to the projection to the continuous spectrum of th Then there exist functions $F(x,y), \Psi(x,y)\in L^{\infty}(\mathbb{T}\times[0,1])$ with such that

Theorems & Definitions (27)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • ...and 17 more