Linear inviscid damping for monotonic shear flow in unbounded domain

Abstract

In this paper, we study the 2-D incompressible Euler equation in unbounded domain $T\times R$, linearized around a class of monotonic shear flow whose derivatives degenerate with same exponentially rate at infinity. We prove the linear inviscid damping with exponential weighted Sobolev initial data.

Our proof includes four parts: limiting absorption principle for Rayleigh equation, space-time estimate, vector field method and semigroup estimate. To seize the degeneracy of the derivatives of the flow, all of our estimates are weighted with the widest range. To handle the lack of compactness for the non-local term, we use blow-up analysis in the proof of limiting absorption principle.