Existence and uniqueness for plane stationary Navier-Stokes flows with compactly supported force

TL;DR

This paper proves the existence and uniqueness of stationary Navier-Stokes solutions in the entire plane with compactly supported force and prescribed limits at infinity, overcoming previous symmetry and smallness restrictions.

Contribution

It introduces new methods to establish existence of solutions with arbitrary compact support force and verifies boundary conditions in multiple scenarios, expanding known solution classes.

Findings

Existence of D-solutions for arbitrary compactly supported force

Verification of boundary conditions in two different scenarios

New estimates controlling velocity differences in Navier-Stokes solutions

Abstract

We study the stationary Navier--Stokes equations in the whole plane with a compactly supported force term and with a prescribed constant spatial limit. Prior to this work, existence of solutions to this problem was only known under special symmetry and smallness assumptions. In the paper we solve the key difficulties in applying Leray's {\it invading domains method} and, as a consequence, prove the existence of $D$-solutions in the whole plane for arbitrary compactly supported force. The boundary condition at infinity are verified in two different scenarios: (I) the~limiting velocity is sufficient large with respect to the external force, (II) both the total integral of force and the~limiting velocity vanish. Hence, our method produces large class of new solutions with prescribed spatial limits. Moreover, we show the uniqueness of $D$-solutions to this problem in a perturbative regime. The main tools here are two new estimates for general Navier-Stokes solutions, which have rather simple forms. They control the difference between mean values of the velocity over two concentric circles in terms of the Dirichlet integral in the annulus between them.