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

Abstract

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 $¥frac{2}{s}$+$¥frac{3}{q}$=1 but not necessarily any differentiability property. The weakest possible boundary conditions, beyond the usual trace theorems, are given by u|_∂Ω=g∈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=divu∈L^s(0,T;L^r(Ω)), where $¥frac{1}{3}$+$¥frac{1}{q}$=$¥frac{1}{r}$.