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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
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