Gevrey stability of Prandtl expansions for 2-dimensional Navier-Stokes flows

David Gérard-Varet, Yasunori Maekawa, Nader Masmoudi

Research output: Contribution to journalArticle

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: uv(t, x,y) = (U E(t, y)+UBL(t, y/√v), 0), 0 < v ≪ 1: We show that if UBL is monotonic and concave in Y = y/√v then uv is stable over some time interval (0, T), T independent of v, 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.

Original languageEnglish (US)
Pages (from-to)2531-2631
Number of pages101
JournalDuke Mathematical Journal
Volume167
Issue number13
DOIs
StatePublished - Sep 1 2018

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Stokes Flow
Navier-Stokes
Gevrey Regularity
Resolvent Estimates
Incompressible Navier-Stokes Equations
Shear Flow
Monotonic
Boundary Layer
Strictly
Regularity
Exponent
Perturbation
Interval
Operator

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Gevrey stability of Prandtl expansions for 2-dimensional Navier-Stokes flows. / Gérard-Varet, David; Maekawa, Yasunori; Masmoudi, Nader.

In: Duke Mathematical Journal, Vol. 167, No. 13, 01.09.2018, p. 2531-2631.

Research output: Contribution to journalArticle

Gérard-Varet, David ; Maekawa, Yasunori ; Masmoudi, Nader. / Gevrey stability of Prandtl expansions for 2-dimensional Navier-Stokes flows. In: Duke Mathematical Journal. 2018 ; Vol. 167, No. 13. pp. 2531-2631.
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