Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 147, pp. 1–10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE AND REGULARITY OF SOLUTIONS FOR STOKES SYSTEMS WITH NON-SMOOTH BOUNDARY DATA IN A POLYHEDRON HOURIA ADJAL, MOHAND MOUSSAOUI, ABDELMALEK ZINE Communicated by Mokhtar Kirane Abstract. The Stokes problem is fundamental in the study of fluid flows. In the case of smooth domains and data, this problem is extensively studied in the literature. But there are only a few results for non-smooth boundary data. In [13], there are some promising results in the 2 dimensional case. The aim of this work is to extend those results to a polyhedron domain with non-regular data. 1. Introduction Let Ω be a convex polyhedron of R 3 with boundary Γ. The steady, creeping flow of an incompressible fluid is governed by Stokes system − div(2ηd(u) − pδ)= f in Ω, div u =0 in Ω, u = g on Γ. (1.1) Where u the velocity field and p the pressure are the unknowns of the problem. f and g are given functions respectively defined on Ω and its boundary Γ, and respectively representing the inertia forces and boundary data. Finally, d(u)= (∇u + ∇ t u)/2 is the strain rate tensor, δ the identity tensor and η the viscosity of the fluid, supposed to be constant (Newtonian fluid). In the bi-dimensional framework, the authors obtained in previous work [13], some promising results on the existence and regularity of the solution to the system (1.1). More precisely, the boundary Γ is supposed to be a set of segments Γ i = ]S i ,S i+1 [, i =1,...,N and the data g |Γi ∈ (H s (Γ i )) 2 with −1/2 <s< +1/2. This work represents a generalization of the results obtained in [13]. It concerns the existence and regularity results of solutions to non-homogeneous Stokes system in a polyhedron with non enough regular data g on the boundary. More precisely, we assume that Ω is a convex polyhedron and it is supposed that its boundary Γ is composed of surfaces F i ,i =1,...,N : Γ= ∪ N i=1 F i , Γ i = ∂F i = ∪ Ni j=1 Γ ij . 2010 Mathematics Subject Classification. 35Q35, 76B03, 76N10. Key words and phrases. Stokes system; polyhedron; non smooth boundary data; existence and regularity. c 2017 Texas State University. Submitted October 24, 2016. Published June 22, 2017. 1