One Navier’s problem for the Brinkman system

Abstract

In this paper we study the Brinkman system and the Darcy-Forchheimer-Brinkman system with the boundary condition of the Navier’s type \( {\textbf{u}}_{{\mathbf {\mathcal {T}}}} = {\textbf{g}}_{{\mathbf {\mathcal {T}}}} \), \(\rho =h\) on \(\partial \Omega \) for a bounded planar domain \(\Omega \) with connected boundary. Solutions are looked for in the Sobolev spaces \(W^{s+1,q}(\Omega ,{\mathbb R}^2)\times W^{s,q}(\Omega )\) and in the Besov spaces \(B_{s+1}^{p,r}(\Omega ,{\mathbb R}^2)\times B_s^{q,r}(\Omega )\). Classical solutions are from the spaces \({\mathcal C}^{k+1,\gamma }(\overline{\Omega },{\mathbb R}^2) \times {\mathcal C}^{k,\gamma }(\overline{\Omega })\). For the Brinkman system we show the unique solvability of the problem. Then we study the Navier problem for the Darcy-Forchheimer-Brinkman system and small boundary conditions.