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On the stability of shear flows in bounded channels, II: non-monotonic shear flows

Alexandru D. Ionescu, Sameer Iyer, Hao Jia

Abstract

We give a proof of linear inviscid damping and vorticity depletion for non-monotonic shear flows with one critical point in a bounded periodic channel. In particular, we obtain quantitative depletion rates for the vorticity function without any symmetry assumptions.

On the stability of shear flows in bounded channels, II: non-monotonic shear flows

Abstract

We give a proof of linear inviscid damping and vorticity depletion for non-monotonic shear flows with one critical point in a bounded periodic channel. In particular, we obtain quantitative depletion rates for the vorticity function without any symmetry assumptions.
Paper Structure (15 sections, 15 theorems, 200 equations)

This paper contains 15 sections, 15 theorems, 200 equations.

Table of Contents

  1. Introduction
  2. Main equations
  3. The main results
  4. Remarks and main ideas of proof
  5. Notations
  6. Spectral property and representation formula
  7. Bounds on the Green's function and modified Green's function
  8. Elementary properties of the standard Green's function
  9. Bounds on the modified Green's function
  10. The limiting absorption principle
  11. The non-degenerate case
  12. The degenerate case $\lambda\in \Sigma_{\delta_0}$
  13. Bounds on $\psi^\iota_{k,\epsilon}$: the non-degenerate case
  14. Bounds on $\psi^\iota_{k,\epsilon}$: the degenerate case
  15. Proof of Theorem \ref{['thm']}

Key Result

Theorem 1.2

Assume that $\omega(t,\cdot)\in C([0,\infty), H^4(\mathbb{T}\times[0,1]))$ with the associated stream function $\psi(t,\cdot)$ is the unique solution to main, with initial data $\omega_0\in H^4(\mathbb{T}\times[0,1])$ satisfying for all $y\in[0,1]$, Then we have the following bounds. (i) Inviscid damping estimates: (ii) Vorticity depletion estimates: there exists a decomposition where for $(x,y

Theorems & Definitions (31)

  • Theorem 1.2
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Corollary 3.3
  • ...and 21 more