Lr -results of the stationary Navier--Stokes equations with nonzero velocity at infinity

We study the stationary motion of an incompressible Navier–Stokes fluid past obstacles in R 3 , subject to the provided boundary velocity u b , external force f = div F , and nonzero constant vector k e 1 at infinity. We first prove that the existence of at least one very weak solution u in L 3 ( Ω ) + L 4 ( Ω ) for an arbitrary large F ∈ L 3 / 2 ( Ω ) + L 2 ( Ω ) provided that the flux of u b on the boundary of each body is sufficiently small with respect to the viscosity ν . Moreover, we establish weak- and strong-regularity results for very weak solutions. Consequently, our existence and regularity results enable us to prove the existence of a weak solution satisfying ∇ u ∈ L r ( Ω ) for a given F ∈ L r ( Ω ) with 3 / 2≤ r ≤2, and a strong solution satisfying ∇ 2 u ∈ L s ( Ω ) for a given f ∈ L s ( Ω ) with 1 ≤6 /5, respectively.