Uniqueness of Some Weak Solutions for 2D Viscous Primitive Equations

First, a new sufficient condition for uniqueness of weak solutions is proved for the system of 2D viscous Primitive Equations. Second, global existence and uniqueness are established for several classes of weak solutions with partial initial regularity, including but not limited to those weak solutions with initial horizontal regularity, rather than vertical regularity. Moreover, new and improved regularity properties of weak solutions and strong solutions to the system of 2D viscous Primitive Equations are obtained. Our results and analyses for the problem with physical boundary conditions can be extended to those with other typical boundary conditions. Most of the results were not available before, even for the periodic case.