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

Abstract

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 ∂ z u ∈ L ∞ ( 0 , T ; L 2 ( Ω ) ) ∩ L 2 ( 0 , T ; H 1 ( Ω ) ) ), 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.