Asymptotic behavior of solutions of the stationary Navier-Stokes equations in an exterior domain

TL;DR

This paper investigates the long-distance decay rates of solutions to the stationary Navier-Stokes equations around obstacles, establishing a minimal exponential decay rate using Carleman estimates.

Contribution

It provides a new analysis of the asymptotic decay behavior of solutions in exterior domains, with a focus on minimal decay rates under certain conditions.

Findings

Nontrivial solutions decay at least as fast as exp(-C t^2 log t) at infinity.

The decay rate is established using Carleman estimates.

Results apply to incompressible fluids around bounded obstacles in ^n for n  2.

Abstract

We study the asymptotic behavior of an incompressible fluid around a bounded obstacle. The problem is modeled by the stationary Navier-Stokes equations in an exterior domain in $\R^n$ with $n\ge 2$. We will show that, under some assumptions, any nontrivial velocity field obeys a minimal decaying rate $\exp(-Ct^2\log t)$ at infinity. Our proof is based on appropriate Carleman estimates.