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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 \(D^{1,q}(\varOmega ,{{\mathbb {R}}}^3)\times L^q(\varOmega )\).

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Correspondence to Dagmar Medková.

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Medková, D. Weak solutions of the Robin problem for the Oseen system. J Elliptic Parabol Equ 5, 189–213 (2019). https://doi.org/10.1007/s41808-019-00038-9

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  • DOI: https://doi.org/10.1007/s41808-019-00038-9

Keywords

  • Oseen system
  • Robin problem
  • Neumann problem

Mathematics Subject Classification

  • 76D10
  • 76D07
  • 35Q35