Asymptotic description of solutions of the exterior Navier Stokes problem in a half space

TL;DR

This paper analyzes the asymptotic behavior of solutions to the Navier-Stokes equations for a small moving body near a wall in a half-space, establishing uniqueness and precise decay rates at infinity.

Contribution

It provides a general proof of uniqueness and derives sharp decay rates for solutions of the exterior Navier-Stokes problem in a half-space with a moving body.

Findings

Solution is unique under general conditions

Decay rate of solutions at infinity is precisely characterized

Results apply to small bodies moving near a wall

Abstract

We consider the problem of a body moving within an incompressible fluid at constant speed parallel to a wall, in an otherwise unbounded domain. This situation is modeled by the incompressible Navier-Stokes equations in an exterior domain in a half space, with appropriate boundary conditions on the wall, the body, and at infinity. We focus on the case where the size of the body is small. We prove in a very general setup that the solution of this problem is unique and we compute a sharp decay rate of the solution far from the moving body and the wall.