## 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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