A well-posed problem for the exterior Stokes equations in two and three dimensions (Q807866)

Let \(\Omega\) be a bounded domain of \(R^ n\) \((n=2\) or \(n=3)\) with boundary \(\Gamma\) and let \(\Omega '\) denote the complement of \({\bar \Omega}\). The authors consider the following exterior problem for the steady-state nonhomogeneous Stokes flow: \[ (*)\quad -\nu \Delta \vec u+\nabla p=\vec f,\quad div \vec u=0\text{ in } \Omega ',\quad \vec u|_{\Gamma}=\vec g, \] with a condition on \(\vec u\) at infinity expressed by \[ \int_{\Omega '}| \vec u|^ 2dx<+\infty,\quad \int_{\Omega '}(1/\omega^ 2)| \vec u|^ 2dx<+\infty. \] Here \(\omega\) is a weight function depending upon the dimension, \(\vec u\) is the velocity, p is the pressure, \(\nu\) is the coefficient of viscosity. Using the weighted Sobolev spaces of Hanouzet (in \(R^ 3)\) and Giroire (in \(R^ 2)\), the authors prove the following result: if \(\Omega \subset R^ n\) has a Lipschitz-continuous boundary \(\Gamma\) that is not necessarily connected, but has no interior connected component and \(\vec f\in (W_ 0^{-1}(\Omega '))^ n\), \(\vec g\in (H^{1/2}(\Omega '))^ n\), then the problem (*) has a unique solution (\(\vec u,p)\in (W^ 1_ 0(\Omega '))^ n\times L^ 2(\Omega ')\), which depends continuously on the data \(\vec f,\vec g\). Moreover, they showed that if the boundary and data are smoother, then so is the solution (\(\vec u,p)\) of the Stokes problem.