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

Abstract

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).