Global well-posedness for axisymmetric Boussinesq system with horizontal viscosity

Abstract: 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 $H<sup>1$-estimate of the velocity via the $L<sup>2$-energy estimate of velocity and the Maximum principle of density. Based on this, we further establish the estimate for the quantity $|\omega(t)|<em>{\sqrt{\mathbb{L}}}:=\sup</em>{2\leq p&lt;\infty}\frac{\norm{\omega(t)}<em>{L<sup>p(\mathbb{R}<sup>3)}}{\sqrt{p}}&lt;\infty$ which implies $|\nabla u(t)|</em>{\mathbb{L}^{{3/2}}:=\sup_{2\leq} p&lt;\infty}\frac{\norm{\nabla u(t)}<em>{L<sup>p(\mathbb{R}<sup>3)}}{p\sqrt{p}}&lt;\infty$. 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</em>{2\leq p&lt;\infty}\int_0^{t\frac{|\nabla} u(\tau)|_{L<sup>p(\mathbb{R}<sup>3)}}{\sqrt{p}}\mathrm{d}\tau&lt;\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.