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

Abstract

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 $\Omega =M\times (-h,h)$, with $M=(0,1)\times (0,1)$. Global well-posedness of strong solutions is established, for any initial data $(v_0,T_0)\in H^1(\Omega )\cap L^{\infty }(\Omega )$ with $({\partial }_z{v}_0,{\nabla }_H{T}_0)\in L^q(\Omega )$ and $v_0\in L_z^1(B_{q,2}^1(M))$, for some $q\in (2,\infty )$, by using delicate energy estimates and maximal regularity estimate in the anisotropic setting.

arXiv admin note: text overlap with arXiv:1703.02512