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Zhang \\cite{CP} proved the regularity of $v$ on $(0,T^\\ast]$ if there exists $p \\in (4, 6)$ such that $$\\int_0^{T^\\ast}\\|v\\cdot e\\|^p_{\\dot{H}^{\\frac{1}{2}+\\frac{2}{p}}}dt < \\infty.$$ J. Chemin, P. Zhang and Z. F. Zhang \\cite{CPZ} extended the range of $p$ to $(4, \\infty)$. In this article we settle the case $p \\in [2, 4]$. 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