On Leray's problem in an infinite-long pipe with the Navier-slip boundary condition

Abstract

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.

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

“…For flows in a nozzle with Navier-slip boundary condition, one can also consider Leray problem where the far field shear flows can also be calculated. The corresponding Leray problem has been studied by [21,24,[28][29][30] and references therein. In the case of three-dimensional pipes with straight outlets, a weak solution of the Navier-Stokes system for arbitrary flux is obtained in [24], which satisfies mixed boundary condition and the far field behavior (2).…”

“…In the case of three-dimensional pipes with straight outlets, a weak solution of the Navier-Stokes system for arbitrary flux is obtained in [24], which satisfies mixed boundary condition and the far field behavior (2). Very recently, Leray problem with Navier boundary condition was solved in [21], as long as the flux Φ is small and the nozzle becomes straight at large distance. They also proved the exponential convergence and regularity of the solution.…”