Weak solutions of the Robin problem for the Oseen system

Abstract

We study the Robin problem for the Oseen system in the Sobolev space $W^{1,q}(\varOmega ;{{\mathbb {R}}}^m)\times L^q(\varOmega )$ on a bounded domain $\varOmega \subset {{\mathbb {R}}}^m$ with Lipschitz boundary for $m=2$ or $m=3$. We prove the unique solvability of the problem for $3/2<q<3$ and $\partial \varOmega$ Lipschitz, and for $1<q<\infty$ and $\partial \varOmega$ 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 $(\mathbf{u},p)\in W^{1,q}_\mathrm{loc}({\overline{\varOmega }} ;{{\mathbb {R}}}^m)\times L^q_\mathrm{loc}({\overline{\varOmega }} )$ 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 $D^{1,q}(\varOmega ,{{\mathbb {R}}}^m)\times L^q(\varOmega )$. Denote by ${\tilde{W}}^{1,q}(\varOmega ;{{\mathbb {R}}}^m)$ the closure of ${{\mathcal {C}}}_c^\infty ({{\mathbb {R}}}^m;{{\mathbb {R}}}^m)$ in $D^{1,q}(\varOmega ,{{\mathbb {R}}}^m)$. If $\varOmega \subset {{\mathbb {R}}}^3$ is an unbounded domain with compact Lipschitz boundary and $3/2<q<3$ then there exists a unique solution of the Robin problem in ${\tilde{W}}^{1,q}(\varOmega ,{{\mathbb {R}}}^3)\times L^q(\varOmega )$. We characterize all solutions of the problem in $D^{1,q}(\varOmega ,{{\mathbb {R}}}^3)\times L^q(\varOmega )$.