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.*