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Instability of shear layers and Prandtl's boundary layers

Dongfen Bian, Emmanuel Grenier

Abstract

This paper is devoted to the study of the nonlinear instability of shear layers and of Prandtl's boundary layers, for the incompressible Navier Stokes equations. We prove that generic shear layers are nonlinearly unstable provided the Reynolds number is large enough, or equivalently provided the viscosity is small enough. We also prove that, generically, Prandtl's boundary layer analysis fails for initial data with Sobolev regularity. In both cases we give an accurate description of the first instability which arises. In some cases a secondary instability appears, leading to several sublayers and to an unexpected complexity of the flow.

Instability of shear layers and Prandtl's boundary layers

Abstract

This paper is devoted to the study of the nonlinear instability of shear layers and of Prandtl's boundary layers, for the incompressible Navier Stokes equations. We prove that generic shear layers are nonlinearly unstable provided the Reynolds number is large enough, or equivalently provided the viscosity is small enough. We also prove that, generically, Prandtl's boundary layer analysis fails for initial data with Sobolev regularity. In both cases we give an accurate description of the first instability which arises. In some cases a secondary instability appears, leading to several sublayers and to an unexpected complexity of the flow.
Paper Structure (24 sections, 15 theorems, 230 equations)

This paper contains 24 sections, 15 theorems, 230 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Orr Sommerfeld equations
  4. Linear instability
  5. Functional spaces
  6. Inverse of Orr Sommerfeld operator
  7. Description of the Green function
  8. Inverse when $| c - c(\alpha,\nu) | \gtrsim \nu^{1/4}$
  9. Inverse for $c$ close to $c(\alpha,\nu)$
  10. Inverse for $c = c(\alpha,\nu)$
  11. "Slow" instabilities
  12. Rewriting Navier Stokes equations
  13. Structure of the instability
  14. Proof of Theorem \ref{['theogeneral']}
  15. "Fast" instabilities
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $U_s(0,y)$ be a smooth profile such that $U_s(0,0) = 0$, $\partial_y U_s(0,0) \ne 0$, such that $U_s(0,y)$ converges exponentially fast to some positive constant $U_+$ and such that $\partial_y^2 U_s(0,y)$ converges exponentially fast to $0$ as $y$ goes to $+ \infty$. Let $U^\nu$ is defined by ( with $V^\nu = 0$ at $y = 0$, a sequence of times $T^\nu$, and a constant $\sigma > 0$ such that an

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Lemma 2.4
  • proof
  • ...and 17 more