On the Stokes problem in Lipschitz domains
The authors consider the Stokes problem in a bounded domain \(\Omega \subset \mathbb{R}^ n (n \geq 2)\), i.e. \[ - \Delta u + \nabla p = f, \text{ div} u = g\quad\text{in } \Omega,\quad u = \varphi\quad\text{on } \partial \Omega \tag{*} \] where \(\partial \Omega\) is only assumed to be Lipschitz. Their main result is the following: let \(1 < q < \infty\); if the Lipschitz constant of \(\partial \Omega\) is sufficiently small, then for each given \(f \in W^{-1,q} (\Omega)^ n\), \(g \in L^ q (\Omega)\) and \(\varphi \in W^{1 - {1 \over q}, q} (\Omega)^ n\) satisfying \(\int_ \Omega gdx = \int_{\partial \Omega} \varphi \cdot N do\) there exists a unique solution \((u,p) \in W^{1,q} (\Omega)^ n \times L^ q (\Omega)\) of \((*)\) such that \(\int_ \Omega pdx = 0\). Moreover \((u,p)\) satisfies the estimate \[ \| u \|_{W^{1,q} (\Omega)^ n} + \| p \|_{L^ q (\Omega)} \leq c \biggl( \| f \|_{W^{- 1,q} (\Omega)^ n} + \| g \|_{L^ q (\Omega)} + \| \varphi \|_{W^{1 - {1 \over q}, q} (\Omega)^ n} \biggr). \] The proof uses localization techniques and suitable a-priori estimates for solutions of the Stokes equations on \(\mathbb{R}^ n\) or on a ``bended'' halfspace.
scientific article; zbMATH DE number 711495
scientific article