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Asymptotic stability of shear flows for 2D Euler equations at Yudovich regularity

Dengjun Guo, Xiaoyutao Luo

Abstract

The nonlinear asymptotic stability of shear flows in the 2D Euler equations has traditionally been linked to inviscid damping in the periodic setting. Since Gevrey regularity is required to suppress the ``echo'' phenomenon, asymptotic stability is known to be impossible in Sobolev spaces. In this paper, we identify a distinct stabilizing mechanism available in the infinite channel: the advection of vorticity to spatial infinity. We establish nonlinear asymptotic stability for the 2D Euler equations in the infinite channel $\mathbb{R}\times[0,1]$ at the minimal regularity of the Yudovich class ($L^{\infty}$ vorticity). Specifically, for a class of non-negative shear flows with a curvature bound, any $L^\infty$-small, compactly supported vorticity perturbation leads to decay on compact subsets and weak convergence to zero.

Asymptotic stability of shear flows for 2D Euler equations at Yudovich regularity

Abstract

The nonlinear asymptotic stability of shear flows in the 2D Euler equations has traditionally been linked to inviscid damping in the periodic setting. Since Gevrey regularity is required to suppress the ``echo'' phenomenon, asymptotic stability is known to be impossible in Sobolev spaces. In this paper, we identify a distinct stabilizing mechanism available in the infinite channel: the advection of vorticity to spatial infinity. We establish nonlinear asymptotic stability for the 2D Euler equations in the infinite channel at the minimal regularity of the Yudovich class ( vorticity). Specifically, for a class of non-negative shear flows with a curvature bound, any -small, compactly supported vorticity perturbation leads to decay on compact subsets and weak convergence to zero.
Paper Structure (23 sections, 17 theorems, 108 equations, 1 table)

This paper contains 23 sections, 17 theorems, 108 equations, 1 table.

Table of Contents

  1. Introduction
  2. Main result
  3. Exponential decay for non-stagnant flow
  4. Loss of compactness
  5. Comparison of stability results
  6. The asymmetry and challenge
  7. The phase mixing paradigm
  8. Orbital vs. asymptotic stability
  9. Methodology and outline
  10. Yudovich solutions on the infinite channel
  11. The Green function and Biot-Savart law
  12. The Yudovich theory in the infinite channel
  13. A maximum principle
  14. Renormalization and conserved quantities
  15. Renormalization identity
  16. ...and 8 more sections

Key Result

Theorem 1.1

There exists a universal constant $C_*>0$ such that for any shear flow $f \in \mathcal{F}$, eq:eu_eq is asymptotically stable in the infinite channel $\Omega = \mathbb{R} \times [0,1]$ with respect to compactly supported Yudovich perturbations. More precisely, for any $f \in \mathcal{F}$, there exis the unique Yudovich solutionDespite the unboundedness of $\Omega$ and the non-decaying background s

Theorems & Definitions (36)

  • Theorem 1.1
  • Remark 1.2: Stability mechanism
  • Remark 1.3: Necessity of no stagnation
  • Remark 1.4: The applicability
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Lemma 2.1
  • ...and 26 more