On the stationary Navier-Stokes equations in the half-plane

TL;DR

This paper proves the existence of strong and weak solutions to the stationary Navier-Stokes equations in the half-plane with boundary data near Jeffery-Hamel solutions, addressing boundary compatibility conditions.

Contribution

It establishes existence results for strong and weak solutions under boundary perturbations close to Jeffery-Hamel solutions, including compatibility conditions.

Findings

Existence of strong solutions for boundary data near Jeffery-Hamel solutions.

Existence of weak solutions and weak-strong uniqueness for small data.

Identification of boundary integral conditions related to flux and asymmetry.

Abstract

We consider the stationary incompressible Navier-Stokes equation in the half-plane with inhomogeneous boundary condition. We prove existence of strong solutions for boundary data close to any Jeffery-Hamel solution with small flux evaluated on the boundary. The perturbation of the Jeffery-Hamel solution on the boundary has to satisfy a nonlinear compatibility condition which corresponds to the integral of the velocity field on the boundary. The first component of this integral is the flux which is an invariant quantity, but the second, called the asymmetry, is not invariant, which leads to one compatibility condition. Finally, we prove existence of weak solutions, as well as weak-strong uniqueness for small data.