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arXiv · 1809.01729

On the forced Euler and Navier-Stokes equations: Linear damping and modified scattering

Christian Zillinger
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Abstract

We study the asymptotic behavior of the forced linear Euler and nonlinear Navier-Stokes equations close to Couette flow in a periodic channel. As our main result we show that for smooth time-periodic forcing linear inviscid damping persists, i.e. the velocity field (weakly) asymptotically converges. However, stability and scattering to the transport problem fail in $H^{s}, s>-1$. We further show that this behavior is consistent with the nonlinear Euler equations and that a similar result also holds for the nonlinear Navier-Stokes equations. Hence, these results provide an indication that nonlinear inviscid damping may still hold in Sobolev regularity in the above sense despite the Gevrey regularity instability results of [Deng-Masmoudi 2018].

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Christian Zillinger. 2018-09-05. On the forced Euler and Navier-Stokes equations: Linear damping and modified scattering. https://doi.org/10.1007/s00021-019-0452-3

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