= Linear inviscid damping in unbounded Couette flow
{title2=$\psi=O(t^{-2}),\quad u=O(t^{-1}),\quad v=O(t^{-2})$}
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.
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