On the stationary Navier–Stokes equations in exterior domains

This paper is concerned with the existence and uniqueness questions on weak solutions of the stationary Navier–Stokes equations in an exterior domain Ω in R 3 , where the external force is given by div F with F = F ( x ) = ( F j i ( x ) ) i , j = 1 , 2 , 3 . First, we prove the existence and uniqueness of a weak solution for F ∈ L 3 / 2 , ∞ ( Ω ) ∩ L p , q ( Ω ) with 3 / 2 < p < 3 and 1 ≤ q ≤ ∞ provided ‖ F ‖ L 3 / 2 , ∞ ( Ω ) is sufficiently small. Here L p , q ( Ω ) denotes the well-known Lorentz space. We next show that weak solutions satisfying the energy inequality are unique for F ∈ L 3 / 2 , ∞ ( Ω ) ∩ L 2 ( Ω ) under the same smallness condition on ‖ F ‖ L 3 / 2 , ∞ ( Ω ) . 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 F in a very large class, for instance, in L 3 / 2 ( Ω ) + L 2 ( Ω ) , which generalizes Leray’s existence result.