An L q ‐theory for weak solutions of the stokes system in exterior domains

We introduce a new concept for weak solutions in L q ‐spaces, 1 < q < ∞, of the Stokes system in an exterior domain Ω ⊂ ℝ n , n ⩾ 2. Defining the variational formulation in the homogeneous Sobolev space \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop H\limits^.{_{0}}^{1,q} (\Omega )^n = \{ u \in L_{1{\rm oc}}^q (\overline \Omega )^n;\nabla u \in L^q (\Omega )^{n^2 },u\left| {_{\partial \Omega } = 0} \right.\},$\end{document} we prove existence and uniqueness of weak solutions for an arbitrary external force and a prescribed divergence g = div u . On the other hand, solutions in the sense of distributions which are defined by taking test functions only in C   0 ∞ (Ω) n are not unique if q > n /( n −1). In this case, a hidden boundary condition related to the force exerted on the body may be imposed to single out a unique solution.