Global Well‐Posedness of the Three‐Dimensional Primitive Equations with Only Horizontal Viscosity and Diffusion

Abstract

In this paper, we consider the initial boundary value problem of the three‐dimensional primitive equations for planetary oceanic and atmospheric dynamics with only horizontal eddy viscosity in the horizontal momentum equations and only horizontal diffusion in the temperature equation. Global well‐posedness of the strong solution is established for any H 2 initial data. An N ‐dimensional logarithmic Sobolev embedding inequality, which bounds the L ∞ ‐norm in terms of the L q ‐norms up to a logarithm of the L p ‐norm for p > N of the first‐order derivatives, and a system version of the classic Grönwall inequality are exploited to establish the required a~priori H 2 estimates for global regularity.© 2016 Wiley Periodicals, Inc.