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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.
}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/537.html} }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.