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 $\mathrm{\Omega }\subset {\mathbf{R}}^3$ in a solution class $L^s(0,T;L^q(\mathrm{\Omega }\left)\right)$ 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 $u{|}_{\partial \mathrm{\Omega }}=g\in L^s(0,T;W^{-1/q,q}(\partial \mathrm{\Omega }\left)\right)$, and will be made precise in this paper. Moreover, we suppose the weakest possible divergence condition $k=÷u\in L^s(0,T;L^r(\mathrm{\Omega }\left)\right)$, where $\frac{1}{3}+\frac{1}{q}=\frac{1}{r}$.