Very weak solutions of the Navier-Stokes equations in exterior domains with nonhomogeneous data

We investigate the nonstationary Navier-Stokes equations for an exterior domain Ω ⊂ R³ in a solution class L^s(0,T; L ^q(Ω)) of very low regularity in space and time, satisfying Serrin's condition 2/s + 3/q = 1 but not necessarily any differentiability property. The weakest possible boundary conditions, beyond the usual trace theorems, are given by u|_∂Ω = g ε L^s(0,T; W^-1/q,q(∂Ω)), and will be made precise in this paper. Moreover, we suppose the weakest possible divergence condition k = div u ε L^s (0,T; L^r(Ω)), where 1/3 + 1/q = 1/r.