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

TL;DR

This paper investigates the existence and behavior of steady incompressible Navier-Stokes solutions in an infinite pipe with Navier-slip boundary conditions, introducing new methods to handle boundary complexities and asymptotic analysis.

Contribution

It extends Leray's problem to Navier-slip boundary conditions, developing novel techniques for boundary term management and asymptotic profile construction.

Findings

Established existence of solutions with Navier-slip boundary conditions.

Constructed a Poiseuille flow as the asymptotic profile at infinity.

Developed energy estimates involving gradient and stress tensor.

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.