Guillén-González, F.; Rodríguez-Bellido, M. A.; Rojas-Medar, M. A. Hydrostatic Stokes equations with non-smooth data for mixed boundary conditions. (English) Zbl 1062.35064 Ann. Inst. Henri Poincaré, Anal. Non Linéaire 21, No. 6, 807-826 (2004). Summary: We 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 C. Conca [Rev. Mat. Apl. 10, 115–122 (1989; Zbl 0702.35202)] imposing non-smooth Dirichlet boundary conditions.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 M. Ziane [Appl. Anal. 58, 263–292 (1995; Zbl 0837.35030)]). 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 F. Guillén-González, N. Masmoudi and M. A. Rodríguez-Bellido [Differential Integral Equations 14, 1381–1408 (2001; Zbl 1161.76454)]. Cited in 2 Documents MSC: 35Q30 Navier-Stokes equations 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 35D05 Existence of generalized solutions of PDE (MSC2000) 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35B65 Smoothness and regularity of solutions to PDEs Keywords:Non-smooth boundary data; Transposition method; primitive equations; strong regularity; existence; uniqueness; very weak solution; non-stationary hydrostatic Navier-Stokes equations Citations:Zbl 0702.35202; Zbl 0837.35030; Zbl 1161.76454 PDFBibTeX XMLCite \textit{F. Guillén-González} et al., Ann. Inst. Henri Poincaré, Anal. Non Linéaire 21, No. 6, 807--826 (2004; Zbl 1062.35064) Full Text: DOI Numdam References: [1] Amrouche, C.; Girault, V., Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czechoslovak Math. 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