Global well-posedness for axisymmetric Boussinesq system with horizontal viscosity

In this paper, we are concerned with the tridimensional anisotropic Boussinesq equations which can be described by {equation*} {{array}{ll} (\partial_{t}+u\cdot\nabla)u-\kappa\Delta_{h} u+\nabla \Pi=\rho e_{3},\quad(t,x)\in\mathbb{R}^{+}\times\mathbb{R}^{3}, (\partial_{t}+u\cdot\nabla)\rho=0, \text{div}u=0. {array}. {equation*} 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 $\|\nabla u(t)\|_{\mathbb{L}^{3/2}}:=\sup_{2\leq p<\infty}\frac{\norm{\nabla u(t)}_{L^p(\mathbb{R}^3)}}{p\sqrt{p}}<\infty$. 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.