On the uniqueness and regularity of the Primitive Equations imposing additional anisotropic regularity

In this note, we prove that given u a weak solution of the Primitive Equations, imposing an additional condition on the vertical derivative of the velocity u (concretely partial derivative(Z)u epsilon L-infinity(0, T; L-2(Omega)) boolean AND L-2 (0, T; H-1(Omega))), then two different results hold; namely, uniqueness of weak solution (any weak solution associated to the same data that u must coincide with u) and global in time strong regularity for u (without "smallness assumptions" on the data). Both results are proved when either Dirichlet or Robin type conditions on the bottom are considered. In the last case, a domain with a strictly bounded from below depth has to be imposed, even for the uniqueness result. (c) 2005 Elsevier Ltd. All rights reserved.