One problem of the Navier type for the Stokes system in planar domains

We study the problem −Δu+∇ρ=F−Δu+∇ρ=F, ∇⋅u=G∇⋅u=G in Ω, u⋅τ=gu⋅τ=g, ρ=hρ=h on ∂Ω, for a bounded simply connected Lipschitz domain in the plane. For F=0F=0, G=0G=0, g∈Lp(∂Ω)g∈Lp(∂Ω), h∈Lq(∂Ω)h∈Lq(∂Ω) we study a solution in the sense of a nontangential limit. For F∈Ws−1,q(Ω,R2)F∈Ws−1,q(Ω,R2), G∈Ws,q(Ω)G∈Ws,q(Ω), g∈Wt−1/p,p(∂Ω)g∈Wt−1/p,p(∂Ω), h∈Ws−1/q,q(∂Ω)h∈Ws−1/q,q(∂Ω) with t≤s+1t≤s+1 we prove the existence of a unique solution (u,ρ)∈Wt,p(Ω,R2)×Ws,q(Ω)(u,ρ)∈Wt,p(Ω,R2)×Ws,q(Ω). For F∈Bs−1q,r(Ω,R2), G∈Bsq,r(Ω), g∈Bt−1/pp,β(∂Ω), h∈Bs−1/qq,r(∂Ω) with t≤s+1t≤s+1 we prove the existence of a unique solution (u,ρ)∈Btp,β(Ω,R2)×Bsq,r(Ω). For F∈Ck−1,γ(Ω‾,R2), G∈Ck,γ(Ω‾), h∈Ck,γ(∂Ω)h∈Ck,γ(∂Ω), g∈Ck+1,γ(∂Ω)g∈Ck+1,γ(∂Ω) we prove the existence of a unique solution (u,ρ)∈Ck+1,γ(Ω‾,R2)×Ck,γ(Ω‾).