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

Jacob Bedrossian, Nader Masmoudi

Research output: Contribution to journalArticle

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 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.

Original languageEnglish (US)
Pages (from-to)195-300
Number of pages106
JournalPublications Mathématiques de L'Institut des Hautes Scientifiques
Volume122
Issue number1
DOIs
StatePublished - Feb 25 2015

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Couette Flow
Shear Flow
Euler Equations
Asymptotic Stability
Damping
Gevrey Spaces
Converge
Vlasov Equation
Kelvin
Strong Convergence
Vorticity
Velocity Field
Regularity
Perturbation
Class

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.

In: Publications Mathématiques de L'Institut des Hautes Scientifiques, Vol. 122, No. 1, 25.02.2015, p. 195-300.

Research output: Contribution to journalArticle

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