### 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) |
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Pages (from-to) | 52-110 |

Number of pages | 59 |

Journal | Advances in Mathematics |

Volume | 292 |

DOIs | |

State | Published - Apr 9 2016 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

*Advances in Mathematics*,

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