Title: Global well-posedness of the 3D primitive equations with horizontal viscosity and vertical diffusivity,
Abstract: In this paper, we consider the 3D primitive equations of oceanic and
atmospheric dynamics with only horizontal eddy viscosities in the horizontal
momentum equations and only vertical diffusivity in the temperature equation.
Global well-posedness of strong solutions is established for any initial data
such that the initial horizontal velocity $v_0\in H^2(\Omega)$ and the initial
temperature $T_0\in H^1(\Omega)\cap L^\infty(\Omega)$ with $\nabla_HT_0\in
L^q(\Omega)$, for some $q\in(2,\infty)$. Moreover, the strong solutions enjoy
correspondingly more regularities if the initial temperature belongs to
$H^2(\Omega)$. The main difficulties are the absence of the vertical viscosity
and the lack of the horizontal diffusivity, which, interact with each other,
thus causing the "\,mismatching\," of regularities between the horizontal
momentum and temperature equations. To handle this "mismatching" of
regularities, we introduce several auxiliary functions, i.e., $\eta, \theta,
\varphi,$ and $\psi$ in the paper, which are the horizontal curls or some
appropriate combinations of the temperature with the horizontal divergences of
the horizontal velocity $v$ or its vertical derivative $\partial_zv$. To
overcome the difficulties caused by the absence of the horizontal diffusivity,
which leads to the requirement of some $L^1_t(W^{1,\infty}_\textbf{x})$-type a
priori estimates on $v$, we decompose the velocity into the
"temperature-independent" and temperature-dependent parts and deal with them in
different ways, by using the logarithmic Sobolev inequalities of the
Brézis-Gallouet-Wainger and Beale-Kato-Majda types, respectively.
Specifically, a logarithmic Sobolev inequality of the limiting type, introduced
in our previous work [12], is used, and a new logarithmic type Gronwall
inequality is exploited.