Perturbation of the Navier-Stokes flow in an annular domain with the non-vanishing outflow condition

The boundary value problem of the Navier-Stokes equations has been studied so far only under the vanishing outflow condition due to Leray. We consider this problem in an annular do- main D = {x ∈ R 2 ; R1 < |x| outflow. In a previous paper of the first author, an exact solution is obtained for a simple boundary condition of non- vanishing outflow type: u = µ Ri er +bieθ on Γi ,i =1 , 2, where µ, b1 ,b 2 are arbitrary constants. In this paper, we show the existence of solu- tions satisfying the boundary condition: u = { µ Ri + ϕi(θ)}er + {bi + ψi(θ)}eθ on Γi ,i =1 , 2, where ϕi(θ) ,ψ i(θ) are 2π-periodicsmooth function of θ, under some additional condition. Let D be an annular domain in R 2 :