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First, we provide another version of the Liouville theorem of \\cite{kpr15} in the case of zero swirl, where we replaced the Dirichlet integrability condition by mild decay conditions. Then we prove some Liouville theorems under the assumption $\\|\\f{u_r}{r}{\\bf 1}_{\\{u_r< -\\f 1r\\}}\\|_{L^{3/2}(\\mbR^3)}< C_{\\sharp}$ where $C_{\\sharp}$ is a universal constant to be specified. In particular, if $u_r(r,z)\\geq -\\f1r$ for $\\forall (r,z)\\in[0,\\oo)\\times\\mbR$, then ${\\bf u}\\equiv 0$. 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