On the Leray problem for steady flows in two-dimensional infinitely long channels with slip boundary conditions

In this paper, we investigate the Leray problem for steady Navier-Stokes system under full slip boundary conditions in a two dimensional channel with straight outlets. The existence of solutions with arbitrary flux in a general channel with slip boundary conditions is established, which tend to the shear flows at far fields. Furthermore, if the flux is suitably small, the solutions are proved to be unique. One of the crucial ingredients is to construct an appropriate flux carrier and to show a Hardy type inequality for flows with full slip boundary conditions.