On the uniqueness and regularity of the Primitive Equations imposing additional anisotropic regularity Original Research Article

In this note, we prove that given View the MathML sourceu a weak solution of the Primitive Equations, imposing an additional condition on the vertical derivative of the velocity View the MathML sourceu (concretely View the MathML source∂zu∈L∞(0,T;L2(Ω))∩L2(0,T;H1(Ω))), then two different results hold; namely, uniqueness of weak solution (any weak solution associated to the same data that View the MathML sourceu must coincide with View the MathML sourceu) and global in time strong regularity for View the MathML sourceu (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.