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

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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.

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Correspondence to Nader Masmoudi.

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J. Bedrossian was partially supported by NSF Postdoctoral Fellowship in Mathematical Sciences, DMS-1103765.

N. Masmoudi was partially supported by NSF grant DMS-1211806.

The current email and address are jacob@cscamm.umd.edu, Department of Mathematics and the Center for Scientific Computation and Mathematical Modeling, University of Maryland, College Park, MD, USA.

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Bedrossian, J., Masmoudi, N. Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations. Publ.math.IHES 122, 195–300 (2015). https://doi.org/10.1007/s10240-015-0070-4

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