In this paper, we analyse the three-dimensional exterior Stokes problem with the Navier slip boundary conditions, describing the flow of a viscous and incompressible fluid past an obstacle where it is assumed that the fluid may slip at the boundary. Because the flow domain is unbounded, we set the problem in weighted spaces in order to control the behavior at infinity of the solutions. This functional framework also allows to prescribe various behaviors at infinity of the solutions (growth or decay). Existence and uniqueness of solutions are shown in a Hilbert setting which gives the tools for a possible numerical analysis of the problem. Weighted Korn's inequalities are the key point in order to study the variational problem.

We study the three‐dimensional stationary exterior Stokes problem with non standard boundary conditions corresponding to a slip‐without‐friction boundary conditions. Because the flow domain is unbounded, we set the problem in weighted spaces in order to control the behavior at infinity of solutions. This functional framework allows to prescribe various behaviors at infinity of the solutions. The established results are related to the existence and the uniqueness of strong and very weak solutions. Our strategy relies on the fact that due to the boundary conditions, the pressure and the velocity can be decoupled and, as a result, we solve two separate systems to find these quantities.

In the present paper, we study the vector potential problem in exterior domains of R 3 . Our approach is based on the use of weighted spaces in order to describe the behavior of functions at infinity. As a first step of the investigation, we prove important results on the Laplace equation in exterior domains with Dirichlet or Neumann boundary conditions. As a consequence of the obtained results on the vector potential problem, we establish useful results on weighted Sobolev inequalities and Helmholtz decompositions of weighted spaces.