TY - JOUR

T1 - Gevrey stability of Prandtl expansions for 2-dimensional Navier-Stokes flows

AU - Gérard-Varet, David

AU - Maekawa, Yasunori

AU - Masmoudi, Nader

N1 - Funding Information:
Maekawa’s work was partially supported by the Japan Society for the Promotion of Science Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers program “Development of Concentrated Mathematical Center Linking to Wisdom of the Next Generation,” organized by the Mathematical Institute of Tohoku University. Gérard-Varet’s work was partially supported by Agence Nationale de la Recherche grant ANR-13-BS01-0003-01 of project Dyficolti and by Université Sorbonne Paris Cité grant ANR-11-IDEX-005.

PY - 2018/9/1

Y1 - 2018/9/1

N2 - 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.

AB - 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.

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U2 - 10.1215/00127094-2018-0020

DO - 10.1215/00127094-2018-0020

M3 - Article

AN - SCOPUS:85053432389

VL - 167

SP - 2531

EP - 2631

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

SN - 0012-7094

IS - 13

ER -