  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. Our results and
analyses for the problem with phys- ical 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.
