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In this paper, we consider the initial-boundary value problem of the 3Dprimitive equations for planetary oceanic and atmospheric dynamics with onlyhorizontal eddy viscosity in the horizontal momentum equations and onlyhorizontal diffusion in the temperature equation. Global well-posedness ofstrong solution is established for any $H^2$ initial data. An $N$-dimensionallogarithmic Sobolev embedding inequality, which bounds the $L^\infty$ norm interms of the $L^q$ norms up to a logarithm of the $L^p$-norm, for $p>N$, of thefirst order derivatives, and a system version of the classic Gronwallinequality are exploited to establish the required a priori $H^2$ estimates forthe global regularity.

Global Well-posedness of the 3D Primitive Equations with Only Horizontal Viscosity and Diffusion