The main subject of this work is to study the concept of very weak solution for the hydrostatic Stokes system with mixed boundary conditions (non-smooth Neumann conditions on the rigid surface and homogeneous Dirichlet conditions elsewhere on the boundary). In the Stokes framework, this concept has been studied by Conca [Rev. Mat. Apl. 10 (1989)] imposing non-smooth Dirichlet boundary conditions. In this paper, we introduce the dual problem that turns out to be a hydrostatic Stokes system with non-free divergence condition. First, we obtain strong regularity for this dual problem (which can be viewed as a generalisation of the regularity results for the hydrostatic Stokes system with free divergence condition obtained by Ziane [Appl. Anal. 58 (1995)]). Afterwards, we prove existence and uniqueness of very weak solution for the (primal) problem. As a consequence of this result, the existence of strong solution for the non-stationary hydrostatic Navier-Stokes equations is proved, weakening the hypothesis over the time derivative of the wind stress tensor imposed by Guillén-González, Masmoudi and Rodríguez-Bellido [Differential Integral Equations 50 (2001)]. © 2004 Elsevier SAS. All rights reserved.216807826Amrouche, C., Girault, V., Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension (1994) Czechoslovak Math. J., 44 (119), pp. 109-140Azérad, P., Guillén, F., Mathematical justification of the hydrostatic approximation in the Primitive Equations of qeophysical fluid dynamics (2001) SIAM J. Math. Anal., 33 (4), pp. 847-859Besson, O., Laydi, M.R., Some estimates for the anisotropic Navier-Stokes equations and for the hydrostatic approximation (1992) M2AN-Mod. Math. Ana. Nume., 7, pp. 855-865Cattabriga, L., Sur un problema al contorno relativo al sistema di equazioni di Stokes (1961) Rend. Mat. Sem. Univ. Padova, 31, pp. 308-340Chacón, T., Guillén, F., An intrinsic analysis of existence of solutions for the hydrostatic approximation of the Navier-Stokes equations (2000) C. R. Acad. Sci. Paris, Série I, 330, pp. 841-846Conca, C., Stokes equations with non-smooth data (1989) Revista de Matemáticas Aplicadas, 10, pp. 115-122Girault, V., Raviart, P.A., (1986) Finite Element Methods for Navier-Stokes Equations, , Berlin: Springer-VerlagGuillén-González, F., Rodríguez-Bellido, M.A., On the strong solutions of the Primitive Equations in 2D domains (2002) Nonlin. Anal., 50, pp. 621-646Guillén-González, F., Masmoudi, N., Rodríguez-Bellido, M.A., Anisotropic estimates and strong solutions of the Primitive Equations (2001) Differential Integral Equations, 14 (11), pp. 1381-1408Lewandowski, R., (1997) Analyse Mathématique et Océanographie, , MassonLions, J.L., Magenes, E., (1969) Problèmes aux Limites Non Homogènes et Applications, 1. , Paris: DunodLions, J.L., Temam, R., Wang, S., New formulation of the primitive equations of the atmosphere and applications (1992) Nonlinearity, 5, pp. 237-288Lions, J.L., Temam, R., Wang, S., On the equations of the large scale ocean (1992) Nonlinearity, 5, pp. 1007-1053Pedlosky, J., (1987) Geophysical Fluid Dynamics, , Berlin: Springer-VerlagTemam, R., (1977) Navier-Stokes Equations: Theory and Numerical Analysis, , Amsterdam: North HollandZiane, M., Regularity results for Stokes type systems (1995) Appl. Anal., 58, pp. 263-292