Constrained large solutions to Leray's problem in a distorted strip with the Navier-slip boundary condition

TL;DR

This paper addresses the existence, uniqueness, and behavior of solutions to Leray's problem for the stationary Navier-Stokes equations in a 2D distorted strip with Navier-slip boundary conditions, highlighting the influence of friction coefficient and flux size.

Contribution

It introduces new results on large flux solutions in 2D with Navier-slip conditions, especially when the friction coefficient is small, differing from 3D cases.

Findings

Total flux can be large when the friction coefficient is small.

Constants in estimates are independent of the friction coefficient.

The solution's behavior depends on the slip length and boundary friction.

Abstract

In this paper, we will solve the Leray's problem for the stationary Navier-Stokes system in a 2D infinite distorted strip with the Navier-slip boundary condition. The existence, uniqueness, regularity and asymptotic behavior of the solution will be investigated. Moreover, we discuss how the friction coefficient affects the well-posedness of the solution. Due to the validity of the Korn's inequality, all constants in each a priori estimate are independent of the friction coefficient. The main novelty is the total flux of the velocity can be relatively large (proportional to the {\it slip length}) when the friction coefficient of the Navier-slip boundary condition is small, which is essentially different from the 3D case.