Very weak solutions of the stationary Navier-Stokes equations for an incompressible fluid past obstacles

We consider the stationary motion of an incompressible Navier-Stokes fluid past obstacles in R3, subject to the given boundary velocity vb, external force f=divF and nonzero constant vector ke1 at infinity. Our main result is the existence of at least one very weak solution v in ke1+L3(Ω) for arbitrary large F∈L3/2(Ω)+L12/7(Ω) provided that the flux of vb−ke1 on the boundary of each body is sufficiently small with respect to the viscosity ν. The uniqueness of very weak solutions is proved by assuming that F and vb−ke1 are suitably small. Moreover, we establish weak and strong regularity results for very weak solutions. In particular, our existence and regularity results enable us to prove the existence of a weak solution v satisfying ∇v∈L3/2(Ω).