Weak Solution of One Navier’s Problem for the Stokes Resolvent System

Abstract

This paper studies the Stokes resolvent system \(-\Delta \textbf{u}+\lambda \textbf{u}+\nabla \rho =\textbf{f}\), \(\nabla \cdot \textbf{u}=\chi \) in \(\Omega \) with the Navier condition \(\textbf{u}_\textbf{n}=\textbf{g}_\textbf{n}\), \([\partial \textbf{u}/\partial \textbf{n}-\rho \textbf{n}+b\textbf{u}]_\tau =\textbf{h}_\tau \) on \(\partial \Omega \). Here \(\Omega \subset {{\mathbb {R}}}^2\) is a bounded domain with Lipschitz boundary. \(\Omega \) might have holes. First we define and study weak solutions in \(W^{1,2}(\Omega ;{{\mathbb {C}}}^2)\times L^2(\Omega ;{{\mathbb {C}}})\). Using this result we are able to prove the existence of strong solutions of the problem in Sobolev spaces \(W^{s,q}(\Omega ;{{\mathbb {C}}}^2)\times W^{s-1,q}(\Omega ;{{\mathbb {C}}})\), in Besov spaces \(B_s^{q,r}(\Omega ,{{\mathbb {C}}}^2)\times B_{s-1}^{q,r}(\Omega ;{{\mathbb {C}}})\) and classical solutions in the spaces \({{\mathcal {C}}}^{k,\alpha } ({\overline{\Omega }} ;{{\mathbb {C}}}^2)\times {{\mathcal {C}}}^{k-1,\alpha }({\overline{\Omega }} ;{{\mathbb {C}}})\).