TY - JOUR

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

AU - Grenier, Emmanuel

AU - Guo, Yan

AU - Nguyen, Toan T.

N1 - Funding Information:
The authors would like to thank David Gérard-Varet and Mark Williams for their many fruitful discussions and useful comments on an earlier draft of the paper. YG's research is supported in part by NSFC grant 10828103 , NSF grant DMS-1209437 , and a Simon Research Fellowship . TN's is supported in part by NSF grant DMS-1338643 . Guo and Nguyen wish to thank Beijing International Center for Mathematical Research, and Nguyen thanks l'Institut de Mathématiques de Jussieu and ENS Lyon, for their support and hospitality in the summer of 2012 and 2013, during which part of this research was carried out.
Publisher Copyright:
© 2016 Published by Elsevier Inc.

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.

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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 -