perturbation is sufficiently regular. For this special case, a precise description of this is given by the explicit Fourier characterization of the velocity field, namely $$\begin{aligned}
  \tilde{v}(t,k,\eta+kt)= (-i(\eta+kt), ik) \frac{1}{k^{2}+(\eta-kt)^{2}}\tilde{\omega}(0,k,\eta),\end{aligned}$$ where $ t,\eta \in \R, k \in \ZZ$. By Plancherel’s theorem and noting that $(k,\eta)\mapsto (k,\eta+kt)$ is an isometry, we observe the decay and convergence of $\tilde{v}$ as well as its precise dependence on the initial data.

If the underlying profile is not of this special form, the problem is not explicitly solvable anymore and much effort has to be invested to establish similar results. For a discussion of the literature we refer to [@bedrossian2015dynamics] and just briefly mention the following works:

-   In [@bedrossian2015inviscid], nonlinear inviscid damping for Gevrey regular perturbations around Couette flow on $\T \times \R$ is established using methods of pseudo-differential calculus, paraproducts and scattering methods. In particular, their analysis yields a fine description of the nonlinear dynamics and “echos” and has been extended to many further settings such as the 3D or viscous setting with coauthors.

-   In [@Zill3], using similar methods, the second author established linear inviscid damping for, roughly speaking, Bilipschitz flow profiles also in the setting of domains with boundary such as 