On the stationary Navier–Stokes equations in exterior domains

Abstract: This paper is concerned with the existence and uniqueness questions on weak solutions of the stationary Navier–Stokes equations in an exterior domain in , where the external force is given by with . First, we prove the existence and uniqueness of a weak solution for with and provided is sufficiently small. Here denotes the well-known Lorentz space. We next show that weak solutions satisfying the energy inequality are unique for under the same smallness condition on . This result provides a complete answer to the uniqueness question of weak solutions satisfying the energy inequality, the existence of which was proved by Leray in 1933. Finally, we establish the existence of weak solutions for data in a very large class, for instance, in , which generalizes Leray’s existence result. [Copyright &y& Elsevier]