Global well-posedness of the 3D primitive equations with horizontal viscosity and vertical diffusivity II: close to $H^1$ initial data

In this paper, we consider the initial-boundary value problem to the three-dimensional primitive equations for the oceanic and atmospheric dynamics with only horizontal eddy viscosities in the horizontal momentum equations and only vertical diffusivity in the temperature equation in the domain $\mathrm{\Omega }=M\times (-h,h)$, with $M=(0,1)\times (0,1)$. Global well-posedness of strong solutions is established, for any initial data $(v_0,T_0)\in H^1(\mathrm{\Omega })\cap L^{\mathrm{\infty }}(\mathrm{\Omega })$ with $({\mathrm{\partial }}_z{v}_0,{\mathrm{\nabla }}_H{T}_0)\in L^q(\mathrm{\Omega })$ and $v_0\in L_z^1(B_{q,2}^1(M))$, for some $q\in (2,\mathrm{\infty })$, by using delicate energy estimates and maximal regularity estimate in the anisotropic setting.