A Note on the Existence of Solutions to the Oseen System in Lipschitz Domains

Abstract.

We study the boundary-value problem associated with the Oseen system in the exterior of m Lipschitz domains of an euclidean point space $\mathcal{E}_n (n = 2,3).$ We show, among other things, that there are two positive constants $\epsilon$ and α depending on the Lipschitz character of Ω such that: (i) if the boundary datum a belongs to L ^q(∂Ω), with q ∈ [2,+∞), then there exists a solution (u, p), with $\user2{u} \in W^{1/q,q}_{{\text{loc}}}(\Omega),$ and u ∈ L ^∞(Ω) if a ∈ L ^∞(∂Ω), expressed by a simple layer potential plus a linear combination of regular explicit functions; as a consequence, u tends nontangentially to a almost everywhere on ∂Ω; (ii) if a ∈ W ^1-1/q,q(∂Ω), with $q \in [2, 3+\epsilon),$ then ∇u, p ∈ L ^q(Ω) and if a ∈ C ^0,μ(∂Ω), with μ ∈ [0, α), then $\user2{u} \in C^{0,\mu} (\overline{\Omega} );$ also, natural estimates holds.