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Weak solutions of the Robin problem for the Oseen system. (English) Zbl 1418.76026
Summary: We study the Robin problem for the Oseen system in the Sobolev space $$W^{1,q}(\Omega ;\mathbb{R}^m)\times L^q(\Omega )$$ on a bounded domain $$\Omega \subset{{\mathbb{R}}}^m$$ with Lipschitz boundary for $$m=2$$ or $$m=3$$. We prove the unique solvability of the problem for $$3/2<q<3$$ and $$\partial \Omega$$ Lipschitz, and for $$1<q<\infty$$ and $$\partial \Omega$$ of class $${{\mathcal{C}}}^1$$. Then we study the problem on unbounded domains with compact Lipschitz boundary. First we study the problem for the homogeneous Oseen system with $$(\mathbf{u},p)\in W^{1,q}_\text{loc}({\overline{\Omega }} ;{{\mathbb{R}}}^m)\times L^q_\text{loc}({\overline{\Omega }} )$$ and the additional condition $$\mathbf{u}(\mathbf{x})\rightarrow 0$$, $$p(\mathbf{x})\rightarrow 0$$ as $$|\mathbf{x}|\rightarrow \infty$$. Then we study the Robin problem for the non-homogeneous Oseen system in homogeneous Sobolev spaces $$D^{1,q}(\Omega ,{{\mathbb{R}}}^m)\times L^q(\Omega )$$. Denote by $${\tilde{W}}^{1,q}(\Omega ;{{\mathbb{R}}}^m)$$ the closure of $${{\mathcal{C}}}_c^\infty ({{\mathbb{R}}}^m;{{\mathbb{R}}}^m)$$ in $$D^{1,q}(\Omega ,{{\mathbb{R}}}^m)$$. If $$\Omega \subset{{\mathbb{R}}}^3$$ is an unbounded domain with compact Lipschitz boundary and $$3/2<q<3$$ then there exists a unique solution of the Robin problem in $${\tilde{W}}^{1,q}(\Omega ,{{\mathbb{R}}}^3)\times L^q(\Omega )$$. We characterize all solutions of the problem in $$D^{1,q}(\Omega ,{{\mathbb{R}}}^3)\times L^q(\Omega )$$.
##### MSC:
 76D10 Boundary-layer theory, separation and reattachment, higher-order effects 76D07 Stokes and related (Oseen, etc.) flows 35Q35 PDEs in connection with fluid mechanics
##### Keywords:
Oseen system; Robin problem; Neumann problem
Full Text:
##### References:
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