Global $H^2$-Regularity Results of the 3D Primitive Equations of the Ocean

Abstract

In this article, we consider the 3D viscous primitive equations (PEs for brevity) of the ocean under two physically relevant boundary conditions for the $H^1$ and $H^2$ smooth initial data, respectively. The $H^2$ regularity result of the solution for the viscous PEs of the ocean has been unknown since the work by Cao and Titi [3], and Kobelkov [26]. In this article we provide the global $H^2$-regularity results of the solution and its time derivatives for the 3D viscous primitive equations of the ocean by using the $L^6$ estimates developed in [3] and some new energy estimate techniques.