Linear inviscid damping in unbounded Couette flow ($\psi =O(t^{-2}),u=O(t^{-1}),v=O(t^{-2})$)

In unbounded Couette flow with velocity $(y,0)$, linear vorticity is transported exactly as $\omega (x,y,t)={\omega }_0(x-yt,y)$. For a nonzero streamwise Fourier mode $\alpha$ and sufficiently localized smooth initial data, inversion by the one-dimensional modified Helmholtz Green function gives the displayed pointwise decay of the streamfunction and velocity, although vorticity does not decay in magnitude. The mechanism is phase mixing: shearing produces increasingly rapid transverse oscillations, whose elliptic velocity reconstruction suppresses them. Smoothness and decay of the initial streamfunction without derivative control do not alone imply these algebraic rates.