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Flexibility and rigidity for the Couette flow in the infinite channel

Dengjun Guo, Xiaoyutao Luo, Guolin Qin

Abstract

We investigate the existence of stationary and traveling wave solutions to the 2D Euler equations near the Couette flow in the infinite channel $\mathbb{R} \times [-1,1]$. For Sobolev spaces $W^{s,p}$ or Hölder spaces $C^s$, we identify the index $s= 1+ \frac1p $ as the vorticity regularity threshold separating flexibility from rigidity. Specifically, for any $s<1+ \frac1p$ we prove the existence of $C^\infty$ smooth, compactly supported steady states and traveling waves arbitrarily close to the Couette flow in all $W^{s,p}$ and $C^{1-}$. Conversely, we establish the non-existence of such relative equilibria in $ W^{s,p}$ with $s>1+ \frac1p$ or $C^{1+}$. A notable feature of the variational construction is that these flexible solutions belong to every Gevrey class strictly below the analytic threshold.

Flexibility and rigidity for the Couette flow in the infinite channel

Abstract

We investigate the existence of stationary and traveling wave solutions to the 2D Euler equations near the Couette flow in the infinite channel . For Sobolev spaces or Hölder spaces , we identify the index as the vorticity regularity threshold separating flexibility from rigidity. Specifically, for any we prove the existence of smooth, compactly supported steady states and traveling waves arbitrarily close to the Couette flow in all and . Conversely, we establish the non-existence of such relative equilibria in with or . A notable feature of the variational construction is that these flexible solutions belong to every Gevrey class strictly below the analytic threshold.
Paper Structure (27 sections, 23 theorems, 227 equations)

This paper contains 27 sections, 23 theorems, 227 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Existence of relative equilibria
  4. Triviality of relative equilibria
  5. Related results
  6. Main ideas of the proof
  7. Organization of the paper
  8. Preliminaries
  9. Notations
  10. Function spaces
  11. Stream-vorticity formulation and the Green function
  12. Technical lemmas
  13. Existence of steady solutions
  14. Variational Construction of the Smooth Steady State
  15. The penalized energy and the admissible class
  16. ...and 12 more sections

Key Result

Theorem 1.3

Fix $0 < \delta \leq 1$. There exists $\epsilon_\delta>0$ with the following property. There exists a family of functions $\omega_{\epsilon}: \Omega \to \mathbb{R}$ for $0< \epsilon\leq \epsilon_\delta$ such that $\omega_{\epsilon} \in C^\infty_c(\Omega)$ is a steady solution to eq:eu_eq couette wi Moreover, for any $0<\epsilon \le \epsilon_{\delta}$, the following asymptotic estimates hold:

Theorems & Definitions (45)

  • Theorem 1.3: Flexibility
  • Corollary 1.4
  • Remark 1.5
  • Theorem 1.6: Rigidity: Sobolev scale
  • Theorem 1.7: Rigidity: Hölder scale
  • Remark 1.8
  • Definition 2.1: Gevrey spaces
  • Proposition 2.2: Local Gevrey regularity without loss
  • Lemma 2.3
  • Lemma 2.4
  • ...and 35 more