Dirichlet Problem for the Stationary Navier-Stokes System on Lipschitz Domains

We consider the stationary Navier-Stokes system on a bounded Lipschitz domain Ω in R ³ with connected boundary ∂Ω. The main concern is the solvability of the Dirichlet problem with external force and boundary data having minimal regularity. Here L ^q _s+1/q-2(ω) denotes the standard Sobolev space with the pair (s, q) being admissible for the unique solvability in L ^q _s+1/q (ω) of the Stokes system. We show that if 1+s≥2/q in addition, then for any and satisfying the necessary compatibility condition, there exists at least one solution in L ^q _s+1/q (ω) + L ² _1/2 (ω) of the Dirichlet problem and this solution has a complete regularity property. The uniqueness of solutions is also shown under the smallness condition on the corresponding norms of the data.