Asymptotic properties of steady plane solutions of the Navier-Stokes equations in the exterior of a half-space

TL;DR

This paper studies the long-distance behavior of steady two-dimensional Navier-Stokes solutions near a half-space boundary, showing the velocity grows slowly and pressure diminishes at infinity.

Contribution

It extends previous work by analyzing solutions with Navier-slip boundary conditions in a half-space, revealing specific asymptotic growth and decay properties.

Findings

Velocity grows slower than √log r at infinity

Pressure converges to zero along rays from the origin

Results apply to solutions with finite Dirichlet integral

Abstract

Motivated by Gilbarg-Weinberger's early work on asymptotic properties of steady plane solutions of the Navier-Stokes equations on a neighborhood of infinity \cite{GW1978} , we investigate asymptotic properties of steady plane solutions of this system on a half-neighborhood of infinity with finite Dirichlet integral and Navier-slip boundary condition, and obtain that the velocity of the solution grows more slowly than $\sqrt{\log r}$, while the pressure converges to $0$ along each ray passing through the origin.