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Asymptotic linear stability of columnar vortices driven by Coriolis force

Shuang Miao, Siqi Ren, Zhifei Zhang

Abstract

In this paper, we establish the asymptotic linear stability of a class of Coriolis-driven columnar vortices for the 3-D axisymmetric Euler equations. This result represents a critical step toward proving the nonlinear asymptotic stability of such vortices. The key and widely applicable strategy is to construct a distorted Fourier basis, which is achieved by solving a two-parameter $(c, ξ)$-dependent Schrödinger equation associated with the linearized operator of the system. To capture the precise asymptotic behavior of the solution, we decompose the $c-ξ$ plane into distinct regions, with the partitioning guided by the leading-order profiles of the Schrödinger equation across different parameter regimes.

Asymptotic linear stability of columnar vortices driven by Coriolis force

Abstract

In this paper, we establish the asymptotic linear stability of a class of Coriolis-driven columnar vortices for the 3-D axisymmetric Euler equations. This result represents a critical step toward proving the nonlinear asymptotic stability of such vortices. The key and widely applicable strategy is to construct a distorted Fourier basis, which is achieved by solving a two-parameter -dependent Schrödinger equation associated with the linearized operator of the system. To capture the precise asymptotic behavior of the solution, we decompose the plane into distinct regions, with the partitioning guided by the leading-order profiles of the Schrödinger equation across different parameter regimes.
Paper Structure (33 sections, 48 theorems, 899 equations, 1 figure)

This paper contains 33 sections, 48 theorems, 899 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main result
  3. Reformulation of the linearized system
  4. Distorted Fourier transform
  5. Overview
  6. Construction of distorted Fourier basis
  7. The resolvent estimate
  8. Notations
  9. Proof of main result assuming estimates on the resolvent kernel
  10. Spectral calculus
  11. The limiting absorption
  12. Proof of Theorem \ref{['Thm:dipersive-decay']}
  13. Revisiting the homogenous case
  14. Reformulation of Proposition \ref{['lem:Boundness-K']}
  15. Behavior of Schrödinger equation near the origin
  16. ...and 18 more sections

Key Result

Theorem 1.1

Let $m,n\in\mathbb{N}$ and $0<\delta\ll 1$ be a fixed number. Under the assumptions (A1)-(A3), the solution of the linearized Euler-Coriolis system eq:vel-linearize-gernal0 admits the following uniform decay bound for $t\gtrsim 1$, Here the norm of the weighted Sobolev space $W^{k,1}_{\omega}$ is defined by

Figures (1)

  • Figure 1:

Theorems & Definitions (105)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Lemma 2.5
  • proof
  • ...and 95 more