### Abstract

In this paper, we prove the spectral instability of general symmetric shear flows of the incompressible Navier-Stokes equations at a high Reynolds number in a two-dimensional channel. This includes shear flows that are spectrally stable to the corresponding Euler equations, and thus for the first time, provides a complete mathematical proof of the viscous destabilization phenomenon, pointed out by Heisenberg (1924) [5], C.C. Lin (1944) [9] and Tollmien (1947) [17], among others. Precisely, we construct exact unstable eigenvalues and eigenfunctions of the linearized Navier-Stokes equations around symmetric shear flows, showing that the solution could grow slowly at the rate of et/αR, where R is the sufficiently large Reynolds number and α is the small spatial frequency that remains between lower and upper marginal stability curves: α_{low}(R)≈R^{-1/7} and α_{up}(R)≈R^{-1/11}. We introduce a new, operator-based approach, which avoids to deal with matching inner and outer asymptotic expansions, but instead involves a careful study of singularity in the critical layers by deriving pointwise bounds on the Green function of the corresponding Rayleigh and Airy operators.

Original language | English (US) |
---|---|

Pages (from-to) | 52-110 |

Number of pages | 59 |

Journal | Advances in Mathematics |

Volume | 292 |

DOIs | |

State | Published - Apr 9 2016 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Advances in Mathematics*,

*292*, 52-110. https://doi.org/10.1016/j.aim.2016.01.007

}

*Advances in Mathematics*, vol. 292, pp. 52-110. https://doi.org/10.1016/j.aim.2016.01.007

**Spectral instability of general symmetric shear flows in a two-dimensional channel.** / Grenier, Emmanuel; Guo, Yan; Nguyen, Toan.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Spectral instability of general symmetric shear flows in a two-dimensional channel

AU - Grenier, Emmanuel

AU - Guo, Yan

AU - Nguyen, Toan

PY - 2016/4/9

Y1 - 2016/4/9

N2 - In this paper, we prove the spectral instability of general symmetric shear flows of the incompressible Navier-Stokes equations at a high Reynolds number in a two-dimensional channel. This includes shear flows that are spectrally stable to the corresponding Euler equations, and thus for the first time, provides a complete mathematical proof of the viscous destabilization phenomenon, pointed out by Heisenberg (1924) [5], C.C. Lin (1944) [9] and Tollmien (1947) [17], among others. Precisely, we construct exact unstable eigenvalues and eigenfunctions of the linearized Navier-Stokes equations around symmetric shear flows, showing that the solution could grow slowly at the rate of et/αR, where R is the sufficiently large Reynolds number and α is the small spatial frequency that remains between lower and upper marginal stability curves: αlow(R)≈R-1/7 and αup(R)≈R-1/11. We introduce a new, operator-based approach, which avoids to deal with matching inner and outer asymptotic expansions, but instead involves a careful study of singularity in the critical layers by deriving pointwise bounds on the Green function of the corresponding Rayleigh and Airy operators.

AB - In this paper, we prove the spectral instability of general symmetric shear flows of the incompressible Navier-Stokes equations at a high Reynolds number in a two-dimensional channel. This includes shear flows that are spectrally stable to the corresponding Euler equations, and thus for the first time, provides a complete mathematical proof of the viscous destabilization phenomenon, pointed out by Heisenberg (1924) [5], C.C. Lin (1944) [9] and Tollmien (1947) [17], among others. Precisely, we construct exact unstable eigenvalues and eigenfunctions of the linearized Navier-Stokes equations around symmetric shear flows, showing that the solution could grow slowly at the rate of et/αR, where R is the sufficiently large Reynolds number and α is the small spatial frequency that remains between lower and upper marginal stability curves: αlow(R)≈R-1/7 and αup(R)≈R-1/11. We introduce a new, operator-based approach, which avoids to deal with matching inner and outer asymptotic expansions, but instead involves a careful study of singularity in the critical layers by deriving pointwise bounds on the Green function of the corresponding Rayleigh and Airy operators.

UR - http://www.scopus.com/inward/record.url?scp=84957075471&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84957075471&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2016.01.007

DO - 10.1016/j.aim.2016.01.007

M3 - Article

AN - SCOPUS:84957075471

VL - 292

SP - 52

EP - 110

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -