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 $$ {\textbf{u}}_{{\mathbf {\mathcal {T}}}} = {\textbf{g}}_{{\mathbf {\mathcal {T}}}} $$ u T = g T , $$\rho =h$$ ρ = h on $$\partial \Omega $$ ∂ Ω for a bounded planar domain $$\Omega $$ Ω with connected boundary. Solutions are looked for in the Sobolev spaces $$W^{s+1,q}(\Omega ,{\mathbb R}^2)\times W^{s,q}(\Omega )$$ W s + 1 , q ( Ω , R 2 ) × W s , q ( Ω ) and in the Besov spaces $$B_{s+1}^{p,r}(\Omega ,{\mathbb R}^2)\times B_s^{q,r}(\Omega )$$ B s + 1 p , r ( Ω , R 2 ) × B s q , r ( Ω ) . Classical solutions are from the spaces $${\mathcal C}^{k+1,\gamma }(\overline{\Omega },{\mathbb R}^2) \times {\mathcal C}^{k,\gamma }(\overline{\Omega })$$ C k + 1 , γ ( Ω ¯ , R 2 ) × C k , 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.