The stationary Navier–Stokes equations in 3D exterior domains. An approach in anisotropically weighted [formula omitted] spaces

Abstract

We consider the three-dimensional exterior problem for stationary Navier–Stokes equations. We prove, under assumptions of smallness of the data, existence and uniqueness of solutions. By setting the problem in weighted spaces where the weights reflect the anisotropic decay properties of the fundamental solution of Oseen, we show the better decay of the solutions outside the wake region. Moreover, the solutions we obtained have a finite Dirichlet integral and under additional assumptions on the weights they are also PR -solutions in the sense of Finn [R. Finn, On the exterior stationary problem for the Navier–Stokes equations, and associated perturbation problems, Arch. Ration. Mech. Anal. 19 (1965) 363–406]. The study relies on an L q -theory for 1 < q < ∞ .