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A well-posed problem for the exterior Stokes equations in two and three dimensions. (English) Zbl 0731.35078

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.

MSC:

35Q30 Navier-Stokes equations
35J25 Boundary value problems for second-order elliptic equations
35Dxx Generalized solutions to partial differential equations
35A15 Variational methods applied to PDEs
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