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Bifurcations of viscous boundary layers in the half space

Dongfen Bian, Emmanuel Grenier, Gérard Iooss

Abstract

It is well-established that shear flows are linearly unstable provided the viscosity is small enough, when the horizontal Fourier wave number lies in some interval, between the so-called lower and upper marginally stable curves. In this article, we prove that, under a natural spectral assumption, shear flows undergo a Hopf bifurcation near their upper marginally stable curve. In particular, close to this curve, there exists space periodic traveling waves solutions of the full incompressible Navier-Stokes equations. For the linearized operator, the occurrence of an essential spectrum containing the entire negative real axis causes certain difficulties which are overcome. Moreover, if this Hopf bifurcation is super-critical, these time and space periodic solutions are linearly and nonlinearly asymptotically stable.

Bifurcations of viscous boundary layers in the half space

Abstract

It is well-established that shear flows are linearly unstable provided the viscosity is small enough, when the horizontal Fourier wave number lies in some interval, between the so-called lower and upper marginally stable curves. In this article, we prove that, under a natural spectral assumption, shear flows undergo a Hopf bifurcation near their upper marginally stable curve. In particular, close to this curve, there exists space periodic traveling waves solutions of the full incompressible Navier-Stokes equations. For the linearized operator, the occurrence of an essential spectrum containing the entire negative real axis causes certain difficulties which are overcome. Moreover, if this Hopf bifurcation is super-critical, these time and space periodic solutions are linearly and nonlinearly asymptotically stable.

Paper Structure

This paper contains 41 sections, 24 theorems, 469 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Function spaces
  4. The Helmholtz decomposition in $\mathcal{X}_{\eta}$
  5. Linear operators
  6. Study of $\boldsymbol{L}_{(\nu )}^{(0)}$ and $\boldsymbol{L}_{(\nu )}$
  7. Semi-groups
  8. Study of $e^{\boldsymbol{L}_{(\nu )}t}$ in $\mathring{L}^{\infty }$ and in $\mathring{L}_{\eta}^{\infty }$
  9. Study of the semi-group $e^{\boldsymbol{L}_{(\nu )}t}$ in $\mathring{L}_{\eta}^{2}$
  10. Study of the quadratic term
  11. Nonlinear stability of $U$
  12. Study of the bifurcation
  13. Setup
  14. Study of the inverse of ${\bf L}_0$
  15. The eigenvectors $\zeta$ and $\zeta^\star$
  16. ...and 26 more sections

Key Result

Theorem 1

Let us assume (A1), (A2) and (A3). Let $\nu_0 > 0$ and let $\alpha = \alpha_+(\nu_0)$, let us consider perturbations which are $2 \pi / \alpha$ periodic in $x$. Then, at $\nu=\nu_0$, the system undergoes a Hopf bifurcation:

Figures (4)

  • Figure 1: Stability of shear flows: horizontally, the Reynolds number (inverse of $\nu$), vertically the horizontal wave length $\alpha$ of the perturbation. In grey, the unstable area. Top sub-figure: Euler-stable profile. Bottom sub-figure: Euler-unstable profile. From Landau.
  • Figure 2: a) Location of the spectrum of $\boldsymbol{L}_{(\nu)}$ inside the region bounded by dashed line b) Location of the essential spectrum in the hatched region $\Sigma_{U_+}$ and on half left real line.
  • Figure 3: Contour $\Gamma$ for the estimate of the semi-group used in Lemma \ref{['lem:estimsemigroup poids']}.
  • Figure 4: Hatched region $\Sigma_{U_+,\omega}$ and negative real line containing the spectrum of $\nu\Pi\Delta-(U_{+}+\frac{\omega }{\alpha })\frac{\partial }{\partial \xi }$ and the essential spectrum of $\boldsymbol{L}_{\nu,\omega}$.

Theorems & Definitions (39)

  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Remark 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • proof
  • ...and 29 more