### Abstract

We prove asymptotic stability of shear flows close to the planar Couette flow in the 2D inviscid Euler equations on T×R. That is, given an initial perturbation of the Couette flow small in a suitable regularity class, specifically Gevrey space of class smaller than 2, the velocity converges strongly in L^{2} to a shear flow which is also close to the Couette flow. The vorticity is asymptotically driven to small scales by a linear evolution and weakly converges as t→±∞. The strong convergence of the velocity field is sometimes referred to as inviscid damping, due to the relationship with Landau damping in the Vlasov equations. This convergence was formally derived at the linear level by Kelvin in 1887 and it occurs at an algebraic rate first computed by Orr in 1907; our work appears to be the first rigorous confirmation of this behavior on the nonlinear level.

Original language | English (US) |
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Pages (from-to) | 195-300 |

Number of pages | 106 |

Journal | Publications Mathématiques de L'Institut des Hautes Scientifiques |

Volume | 122 |

Issue number | 1 |

DOIs | |

State | Published - Feb 25 2015 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations.** / Bedrossian, Jacob; Masmoudi, Nader.

Research output: Contribution to journal › Article

*Publications Mathématiques de L'Institut des Hautes Scientifiques*, vol. 122, no. 1, pp. 195-300. https://doi.org/10.1007/s10240-015-0070-4

}

TY - JOUR

T1 - Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations

AU - Bedrossian, Jacob

AU - Masmoudi, Nader

PY - 2015/2/25

Y1 - 2015/2/25

N2 - We prove asymptotic stability of shear flows close to the planar Couette flow in the 2D inviscid Euler equations on T×R. That is, given an initial perturbation of the Couette flow small in a suitable regularity class, specifically Gevrey space of class smaller than 2, the velocity converges strongly in L2 to a shear flow which is also close to the Couette flow. The vorticity is asymptotically driven to small scales by a linear evolution and weakly converges as t→±∞. The strong convergence of the velocity field is sometimes referred to as inviscid damping, due to the relationship with Landau damping in the Vlasov equations. This convergence was formally derived at the linear level by Kelvin in 1887 and it occurs at an algebraic rate first computed by Orr in 1907; our work appears to be the first rigorous confirmation of this behavior on the nonlinear level.

AB - We prove asymptotic stability of shear flows close to the planar Couette flow in the 2D inviscid Euler equations on T×R. That is, given an initial perturbation of the Couette flow small in a suitable regularity class, specifically Gevrey space of class smaller than 2, the velocity converges strongly in L2 to a shear flow which is also close to the Couette flow. The vorticity is asymptotically driven to small scales by a linear evolution and weakly converges as t→±∞. The strong convergence of the velocity field is sometimes referred to as inviscid damping, due to the relationship with Landau damping in the Vlasov equations. This convergence was formally derived at the linear level by Kelvin in 1887 and it occurs at an algebraic rate first computed by Orr in 1907; our work appears to be the first rigorous confirmation of this behavior on the nonlinear level.

UR - http://www.scopus.com/inward/record.url?scp=84945461569&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84945461569&partnerID=8YFLogxK

U2 - 10.1007/s10240-015-0070-4

DO - 10.1007/s10240-015-0070-4

M3 - Article

AN - SCOPUS:84945461569

VL - 122

SP - 195

EP - 300

JO - Publications Mathematiques de l'Institut des Hautes Etudes Scientifiques

JF - Publications Mathematiques de l'Institut des Hautes Etudes Scientifiques

SN - 0073-8301

IS - 1

ER -