An Lq-theory for weak solutions of the stokes system in exterior domains

✍ Scribed by R. Farwig; C. G. Simader; H. Sohr

• Publisher: John Wiley and Sons
• Year: 1993
• Tongue: English
• Weight: 824 KB
• Volume: 16
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/mma.1670161004

✦ Synopsis

Abstract

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(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.