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Linear inviscid damping in Gevrey spaces

Hao Jia

TL;DR

This work establishes linear inviscid damping and scattering for the 2D Euler equations linearized around a broad class of monotone shear flows in a finite channel, within Gevrey spaces. The authors develop a hybrid spectral-Green’s-function framework, introducing a novel renormalization to variables $(z,v)$ and a shift to isolate singularities, then prove Gevrey bounds for generalized eigenfunctions via limiting absorption principles in both $y$ and $v$ coordinates. Under precise spectral and boundary assumptions, they obtain quantitative bounds on the localized stream function and the renormalized vorticity, and demonstrate the existence of a scattering state $f_ extinfty$ with Gevrey-weighted convergence at rate $t^{-1}$. These results provide essential linear theory infrastructure toward proving nonlinear inviscid damping near general monotone shear flows and offer techniques potentially adaptable to broader fluid-structure problems.

Abstract

We prove linear inviscid damping near a general class of monotone shear flows in a finite channel, in Gevrey spaces. It is an essential step towards proving nonlinear inviscid damping for general shear flows that are not close to the Couette flow, which is a major open problem in 2d Euler equations.

Linear inviscid damping in Gevrey spaces

TL;DR

This work establishes linear inviscid damping and scattering for the 2D Euler equations linearized around a broad class of monotone shear flows in a finite channel, within Gevrey spaces. The authors develop a hybrid spectral-Green’s-function framework, introducing a novel renormalization to variables and a shift to isolate singularities, then prove Gevrey bounds for generalized eigenfunctions via limiting absorption principles in both and coordinates. Under precise spectral and boundary assumptions, they obtain quantitative bounds on the localized stream function and the renormalized vorticity, and demonstrate the existence of a scattering state with Gevrey-weighted convergence at rate . These results provide essential linear theory infrastructure toward proving nonlinear inviscid damping near general monotone shear flows and offer techniques potentially adaptable to broader fluid-structure problems.

Abstract

We prove linear inviscid damping near a general class of monotone shear flows in a finite channel, in Gevrey spaces. It is an essential step towards proving nonlinear inviscid damping for general shear flows that are not close to the Couette flow, which is a major open problem in 2d Euler equations.

Paper Structure

This paper contains 18 sections, 12 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Main equations
  3. The linear inviscid damping
  4. Nonlinear inviscid damping and Gevrey regularity
  5. The main results
  6. Main ideas of the proof
  7. Organization and Notations
  8. Representation formula for the stream function $\psi$
  9. The limiting absorption principles
  10. Elementary properties of the Green's function
  11. The limiting absorption principle in the variable $y$
  12. The limiting absorption principles in the variable $v$
  13. Gevrey bounds for generalized eigenfunctions
  14. Proof of the main theorem
  15. Gevrey spaces and Gevrey bounds on the Green's function
  16. ...and 3 more sections

Key Result

Theorem 1.1

Suppose that $\omega_0\in C^{\infty}(\mathbb{T}\times [0,1])$ satisfies ${\rm supp}\,\omega_0\subseteq \mathbb{T}\times[\vartheta_1,1-\vartheta_1]$ for some $\vartheta_1\in(0,\vartheta_0)$. Let $\omega$ be the smooth solution to main with initial data $\omega_0$ and let $\psi$ be the associated stre Define the change of variables and Assume that for some $\lambda\in(0,\infty)$, Then (i) the loc

Theorems & Definitions (31)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 21 more