Under the assumption that the support of the axisymmetric initial data $\\rho_{0}(r,z)$ does not intersect the axis $(Oz)$, we prove the global well-posedness for this system with axisymmetric initial data. We first show the growth of the quantity $\\frac\\rho r$ for large time by taking advantage of characteristic of transport equation. This growing property together with the horizontal smoothing effect enables us to establish $H^1$-estimate of the velocity via the $L^2$-energy estimate of velocity and the Maximum principle of density. Based on this, we further establish the estimate for the quantity $\\|\\omega(t)\\|_{\\sqrt{\\mathbb{L}}}:=\\sup_{2\\leq p<\\infty}\\frac{\\norm{\\omega(t)}_{L^p(\\mathbb{R}^3)}}{\\sqrt{p}}<\\infty$ which implies $\\|\\nabla u(t)\\|_{\\mathbb{L}^{\\frac{3}{2}}}:=\\sup_{2\\leq p<\\infty}\\frac{\\norm{\\nabla u(t)}_{L^p(\\mathbb{R}^3)}}{p\\sqrt{p}}<\\infty$. However, this regularity for the flow admits forbidden singularity since $ \\mathbb{L}$ (see \\eqref{eq-kl} for the definition) seems be the minimum space for the gradient vector field $u(x,t)$ ensuring uniqueness of flow. To bridge this gap, we exploit the space-time estimate about $ \\sup_{2\\leq p<\\infty}\\int_0^t\\frac{\\|\\nabla u(\\tau)\\|_{L^p(\\mathbb{R}^3)}}{\\sqrt{p}}\\mathrm{d}\\tau<\\infty$ by making good use of the horizontal smoothing effect and micro-local techniques. The global well-posedness for the large initial data is achieved by establishing a new type space-time logarithmic inequality.