Linear Inviscid Damping Near Monotone Shear Flows

Abstract

We give an elementary proof of sharp decay rates and the linear inviscid damping near monotone shear flow in a periodic channel, first obtained in [14]. We shall also obtain the precise asymptotics of the solutions, measured in the space L ∞ .

Linear Inviscid Damping in Gevrey Spaces

“…Indeed, Zillinger [25] showed that scattering does not hold in high Sobolev spaces if one does not assume the vorticity to vanish at the boundary. The boundary effect can also be seen clearly in [10] in the main asymptotic term for the stream function. The assumptions on the support of b ′′ is required to preserve the compact support of ω(t) in T × (0, 1).…”

“…In an important work, Wei, Zhang and Zhao [21] obtained the optimal decay estimates for the linearized problem around monotone shear flows, under very general conditions. In [10] the author identified the main term in the asymptotics of the stream function. From the perspective of the linearized problem, the works [10,21] provided a quite satisfactory picture for the linear inviscid damping problem, in Sobolev spaces.…”

“…In [10] the author identified the main term in the asymptotics of the stream function. From the perspective of the linearized problem, the works [10,21] provided a quite satisfactory picture for the linear inviscid damping problem, in Sobolev spaces.…”

Axi‐symmetrization near Point Vortex Solutions for the 2D Euler Equation

“…For example, in [41] Wei-Zhang-Zhao proved sharp decay estimates for a general class of monotone shear flows in a channel. See also a recent refinement [23] where more precise asymptotics for the stream function was obtained. In [4] Bedrossian-Coti Zelati-Vicol established sharp linear inviscid damping around vortices with strictly decreasing profile in the plane.…”

“…The method introduced in [6] for proving nonlinear asymptotic stability is based on the use of time-dependent imbalanced weights, which are designed carefully to control frequency-dependent resonances. In contrast, the linear stability analysis in [4,23,41] is based on the regularity analysis of resolvents, which, through an oscillatory integral, implies decay in time of the stream function. It is not clear at the moment whether the linear decay estimates can be applied to the nonlinear analysis.…”