Global Uniform Boundedness of Solutions to viscous 3D Primitive Equations with Physical Boundary Conditions

Global uniform boundedness of solutions to 3D viscous Primitive equations in a bounded cylindrical domain with physical boundary condition is proved in space $H^m$ for any $m\geqslant2$. A bounded absorbing set for the solutions in $H^m$ is obtained. These results seem rather difficult for the methods recently developed in [8] and [10]. A completely different approach based on hydrostatic helmholtz decomposition is presented, which can also be applied to cases with other different boundary conditions. Several important results about hydrostatic Leray projector are obtained and utilized. These results are expected to be of general interest and will be helpful as well for solving some other problems for 3D viscous Primitive equations which appeared hard previously for the cases with non-periodic boundary conditions (see e.g. [9]).