On Leray’s problem in an infinitely long pipe with the Navier-slip boundary condition

The original Leray’s problem concerns the well-posedness of weak solutions to the steady incompressible Navier-Stokes equations in a distorted pipe, which approach the Poiseuille flow subject to the no-slip boundary condition at spatial infinity. In this paper, the same problem with the Navier-slip boundary condition instead of the no-slip boundary condition, is addressed. Due to the complexity of the boundary condition, some new ideas, presented as follows, are introduced to handle the extra difficulties caused by boundary terms. First, the Poiseuille flow in the semi-infinite straight pipe with the Navier-slip boundary condition will be introduced, which will serve as the asymptotic profile of the solution to the generalized Leray’s problem at spatial infinity. Second, a solenoidal vector function defined in the whole pipe, satisfying the Navier-slip boundary condition, having the designated flux and equalling the Poiseuille flow at a large distance, will be carefully constructed. This plays an important role in reformulating our problem. Third, the energy estimates depend on a combined L2-estimate of the gradient and the stress tensor of the velocity.