One Navier’s problem for the Brinkman system

Abstract

AbstractIn this paper we study the Brinkman system and the Darcy-Forchheimer-Brinkman system with the boundary condition of the Navier’s type$$ {\textbf{u}}_{{\mathbf {\mathcal {T}}}} = {\textbf{g}}_{{\mathbf {\mathcal {T}}}} $$uT=gT,$$\rho =h$$ρ=hon$$\partial \Omega $$∂Ωfor a bounded planar domain$$\Omega $$Ωwith connected boundary. Solutions are looked for in the Sobolev spaces$$W^{s+1,q}(\Omega ,{\mathbb R}^2)\times W^{s,q}(\Omega )$$Ws+1,q(Ω,R2)×Ws,q(Ω)and in the Besov spaces$$B_{s+1}^{p,r}(\Omega ,{\mathbb R}^2)\times B_s^{q,r}(\Omega )$$Bs+1p,r(Ω,R2)×Bsq,r(Ω). Classical solutions are from the spaces$${\mathcal C}^{k+1,\gamma }(\overline{\Omega },{\mathbb R}^2) \times {\mathcal C}^{k,\gamma }(\overline{\Omega })$$Ck+1,γ(Ω¯,R2)×Ck,γ(Ω¯). 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.