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Specifically, for periodic boundary conditions on $[0,L]^2$, and a force $g\\in\\calD(A^{\\frac{\\alpha-1}{2}})$, we show there is a fixed strip about the real time axis on which a uniform bound $|A^{\\alpha}u|< m_\\alpha\\nu\\kappa_0^\\alpha$ holds for each $\\alpha \\in \\bN$. Here $\\nu$ is viscosity, $\\k0=2\\pi/L$, and $m_\\alpha$ is explicitly given in terms of $g$ and $\\alpha$. 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