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

TL;DR

This paper proves the existence and uniqueness of steady Navier-Stokes solutions with slip boundary conditions in 2D channels, using flux carriers and Hardy inequalities, advancing understanding of flow behavior in such geometries.

Contribution

It establishes the existence of solutions with arbitrary flux and their uniqueness for small flux in 2D channels with slip boundaries, introducing flux carriers and Hardy inequalities.

Findings

Existence of solutions with arbitrary flux in 2D channels

Uniqueness of solutions for small flux

Development of Hardy type inequalities for slip boundary flows

Abstract

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.