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On nonlinear instability of Prandtl's boundary layers: the case of Rayleigh's stable shear flows

Emmanuel Grenier, Toan T. Nguyen

Abstract

In 1904, Prandtl introduced his famous boundary layer in order to describe the behavior of solutions of Navier Stokes equations near a boundary as the viscosity goes to $0$. His Ansatz has later been justified for analytic data by R.E. Caflisch and M. Sammartino. In this paper, we prove that his expansion is false, up to $O(ν^{1/4})$ order terms in $L^\infty$ norm, in the case of solutions with Sobolev regularity, even in cases where the Prandlt's equation is well posed in Sobolev spaces. In addition, we also prove that monotonic boundary layer profiles, which are stable when $ν= 0$, are nonlinearly unstable when $ν> 0$, provided $ν$ is small enough, up to $O(ν^{1/4})$ terms in $L^\infty$ norm.

On nonlinear instability of Prandtl's boundary layers: the case of Rayleigh's stable shear flows

Abstract

In 1904, Prandtl introduced his famous boundary layer in order to describe the behavior of solutions of Navier Stokes equations near a boundary as the viscosity goes to . His Ansatz has later been justified for analytic data by R.E. Caflisch and M. Sammartino. In this paper, we prove that his expansion is false, up to order terms in norm, in the case of solutions with Sobolev regularity, even in cases where the Prandlt's equation is well posed in Sobolev spaces. In addition, we also prove that monotonic boundary layer profiles, which are stable when , are nonlinearly unstable when , provided is small enough, up to terms in norm.

Paper Structure

This paper contains 14 sections, 11 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Stable boundary layer profiles
  3. Main results
  4. Boundary layer norms
  5. Linear instability
  6. Spectral instability
  7. Linear estimates
  8. Approximate solutions
  9. Formal construction
  10. Estimates
  11. The remainder
  12. Proof of Theorem \ref{['theo-approx-stable']}
  13. Nonlinear instability
  14. Elliptic estimates

Key Result

Theorem 1.1

Let $U_\mathrm{bl}(t,z)$ be a time-dependent stable boundary layer profile as described in Section sec-BL. Then, for arbitrarily large $s,N$ and arbitrarily small positive $\epsilon$, there exists a sequence of functions $u^\nu$ that solves the Navier-Stokes equations NS1-NS3, with some forcing $f^\ but for time sequences $T^\nu \to 0$, as $\nu \to 0$. Here, $\omega^\nu = \nabla\times u^\nu$ deno

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2: Instability result for stable profiles
  • Theorem 2.1: Spectral instability; GGN3
  • Theorem 2.2
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 7 more