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15 September 2018 Gevrey stability of Prandtl expansions for 2-dimensional Navier–Stokes flows
David Gérard-Varet, Yasunori Maekawa, Nader Masmoudi
Duke Math. J. 167(13): 2531-2631 (15 September 2018). DOI: 10.1215/00127094-2018-0020

Abstract

We investigate the stability of boundary layer solutions of the 2-dimensional incompressible Navier–Stokes equations. We consider shear flow solutions of Prandtl type: uν(t,x,y)=(UE(t,y)+UBL(t,yν),0), 0<ν1. We show that if UBL is monotonic and concave in Y=y/ν, then uν is stable over some time interval (0,T), T independent of ν, under perturbations with Gevrey regularity in x and Sobolev regularity in y. We improve in this way the classical stability results of Sammartino and Caflisch in analytic class (both in x and y). Moreover, in the case where UBL is steady and strictly concave, our Gevrey exponent for stability is optimal. The proof relies on new and sharp resolvent estimates for the linearized Orr–Sommerfeld operator.

Citation

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David Gérard-Varet. Yasunori Maekawa. Nader Masmoudi. "Gevrey stability of Prandtl expansions for 2-dimensional Navier–Stokes flows." Duke Math. J. 167 (13) 2531 - 2631, 15 September 2018. https://doi.org/10.1215/00127094-2018-0020

Information

Received: 22 July 2016; Revised: 26 March 2018; Published: 15 September 2018
First available in Project Euclid: 21 August 2018

zbMATH: 06970974
MathSciNet: MR3855356
Digital Object Identifier: 10.1215/00127094-2018-0020

Subjects:
Primary: 35Q30
Secondary: 35Q35

Keywords: boundary layers , fluid mechanics , Navier–Stokes equations , partial differential equations

Rights: Copyright © 2018 Duke University Press

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Vol.167 • No. 13 • 15 September 2018
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