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

By Jacob Bedrossian, Nader Masmoudi. 2013

308 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Mechanics, Finance

View Paper: https://www.semanticscholar.org/paper/20b43027375825b37c9fefb101f3ad03c93f91d6

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Citations in the corpus

On the stability threshold for the 3D Couette flow in Sobolev regularity (2015, methodology): “The estimates here differ from the analogous estimates employed previously in [9, 11, 7, 8] due to the much lower regularity and the fact that the coefficients are a little smaller here (relative to the primary unknowns).”
Linear inviscid damping and enhanced dissipation for the Kolmogorov flow (2017, background): “Very recently, Ibrahim, Maekawa and Masmoudi [19] proved the same decay rate of the vorticity for the following equation: ∂tω + Lνw = 0, Lν = −ν∆− a cos y∂x(1 + ∆−1), which is the linearized equation of the 2-”
Enhanced Dissipation and Inviscid Damping in the Inviscid Limit of the Navier–Stokes Equations Near the Two Dimensional Couette Flow (2014, background): “In [9] it is shown that this enstrophy flux persists in the Euler equations.”
Enhanced dissipation and Hörmander's hypoellipticity (2021, background): “Notable papers on nonlinear stability of the Couette flow include [10, 11, 12, 34].”