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Under suitable conditions for initial values, we prove the following a priori bound \\[ |v(x, t)| \\le \\frac{C}{r^2} |\\ln r|^{1/2}, \\]where $r \\in (0, 1/2)$ is the distance from $x$ to the z axis, and $C$ is a constant depending only on the initial value.\n  This provides a pointwise upper bound (worst case scenario) for possible singularities while the recent papers \\cite{CSTY2} and \\cite{KNSS} gave a lower bound. 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Navas, Qi S. Zhang, Zhen Lei","submitted_at":"2013-09-25T19:55:31Z","abstract_excerpt":"Let $v$ be the velocity of Leray-Hopf solutions to the axially symmetric three-dimensional Navier-Stokes equations. Under suitable conditions for initial values, we prove the following a priori bound \\[ |v(x, t)| \\le \\frac{C}{r^2} |\\ln r|^{1/2}, \\]where $r \\in (0, 1/2)$ is the distance from $x$ to the z axis, and $C$ is a constant depending only on the initial value.\n  This provides a pointwise upper bound (worst case scenario) for possible singularities while the recent papers \\cite{CSTY2} and \\cite{KNSS} gave a lower bound. 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