Decay estimates for steady solutions of the Navier-Stokes equations in two dimensions in the presence of a wall

TL;DR

This paper establishes decay estimates for steady two-dimensional Navier-Stokes solutions with boundary effects, showing that the vorticity's weighted magnitude remains bounded in a half-plane with small forcing.

Contribution

It provides new decay bounds for stationary Navier-Stokes solutions near a boundary, using a specialized functional framework to extend previous results.

Findings

|xyw(x,y)| is uniformly bounded in the half-plane

Decay estimates hold for solutions with small compactly supported force

The approach complements and extends prior work on boundary decay

Abstract

Let w be the vorticity of a stationary solution of the two-dimensional Navier-Stokes equations with a drift term parallel to the boundary in the half-plane -\infty<x<\infty, y>1, with zero Dirichlet boundary conditions at y=1 and at infinity, and with a small force term of compact support. Then, |xyw(x,y)| is uniformly bounded in the half-plane. The proof is given in a specially adapted functional framework and complements previous work.