One Navier’s problem for the Brinkman system

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