  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 to the Poiseuille flow subject to the no-slip boundary condition at
spacial 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 be served as the
asymptotic profile of the solution to the generalized Leray's problem at
spacial infinity. Second, a solenoidal vector function defined in the whole
pipe, satisfying the Navier-slip boundary condition, having the designated flux
and equalling to the Poiseuille flow at large distance, will be carefully
constructed. This plays an important role in reformulating our problem. Third,
the energy estimates depend on a combined $L^2$ estimate of the gradient and
the stress tensor of the velocity.

Analysis of PDEs