The steady Navier–Stokes problem with the inhomogeneous Navier-type boundary conditions in a 2D multiply-connected bounded domain

Abstract

Abstract We assume that Ω is a bounded domain in ℝ 2 ${{\mathbb {R}}^2}$ with N holes. We study the steady Navier–Stokes problem with inhomogeneous Navier-type boundary conditions in Ω (problem BVP-1) and the two other related problems BVP-2 and BVP-3. We prove the existence of strong solutions of BVP-2 and BVP-3 and the uniqueness of solutions for “small data”. The existence of a strong solution of BVP-1 is proven only if the body force 𝐟 ${\mathbf {f}}$ belongs to a certain set 𝔐 ( ℝ N ) ${\mathfrak {M}({\mathbb {R}}^N)}$ . We show that problem BVP-1 is not solvable for some configurations of given boundary data and body forces.