Well-posedness of the two-dimensional stationary Navier--Stokes equations around a uniform flow

TL;DR

This paper proves the unique existence of solutions to the 2D stationary Navier--Stokes equations around a uniform flow in critical Besov spaces, contrasting previous ill-posedness results around zero.

Contribution

It establishes well-posedness for solutions near a non-zero uniform flow, highlighting the importance of the flow's non-zero condition for regularity.

Findings

Solutions exist uniquely in critical Besov spaces around non-zero flow.

Ill-posedness persists for solutions around zero flow.

Linear part regularity improves with non-zero background flow.

Abstract

In this paper, we consider the solvability of the two-dimensional stationary Navier--Stokes equations on the whole plane $\mathbb{R}^2$. In [6], it was proved that the stationary Navier--Stokes equations on $\mathbb{R}^2$ is ill-posed for solutions around zero. In contrast, considering solutions around the non-zero constant flow, the perturbed system has a better regularity in the linear part, which enables us to prove the unique existence of solutions in the scaling critical spaces of the Besov type.