### 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: u^{v}(t, x,y) = (U ^{E}(t, y)+U^{BL}(t, y/√v), 0), 0 < v ≪ 1: We show that if U^{BL} is monotonic and concave in Y = y/√v then u^{v} 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 U^{BL} 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 language | English (US) |
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Pages (from-to) | 2531-2631 |

Number of pages | 101 |

Journal | Duke Mathematical Journal |

Volume | 167 |

Issue number | 13 |

DOIs | |

State | Published - Sep 1 2018 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Duke Mathematical Journal*,

*167*(13), 2531-2631. https://doi.org/10.1215/00127094-2018-0020

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

Research output: Contribution to journal › Article

*Duke Mathematical Journal*, vol. 167, no. 13, pp. 2531-2631. https://doi.org/10.1215/00127094-2018-0020

}

TY - JOUR

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

AU - Gérard-Varet, David

AU - Maekawa, Yasunori

AU - Masmoudi, Nader

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.

UR - http://www.scopus.com/inward/record.url?scp=85053432389&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85053432389&partnerID=8YFLogxK

U2 - 10.1215/00127094-2018-0020

DO - 10.1215/00127094-2018-0020

M3 - Article

VL - 167

SP - 2531

EP - 2631

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

SN - 0012-7094

IS - 13

ER -