Asymptotics of solutions to the Navier-Stokes system in exterior domains

TL;DR

This paper investigates the long-time behavior of solutions to the Navier-Stokes equations in exterior domains, comparing them with solutions in the whole space, and identifies conditions under which their asymptotics coincide.

Contribution

It provides a detailed analysis of the asymptotic behavior of Navier-Stokes solutions in exterior domains, highlighting cases where solutions behave similarly to those in the whole space.

Findings

Asymptotics coincide for small initial data in weak L^n space.

Asymptotics coincide for certain large initial data.

Long-time behavior in exterior domains matches that in the whole space under specified conditions.

Abstract

We consider the incompressible Navier-Stokes equations with the Dirichlet boundary condition in an exterior domain of $\mathbb{R}^n$ with $n\geq2$. We compare the long-time behaviour of solutions to this initial-boundary value problem with the long-time behaviour of solutions of the analogous Cauchy problem in the whole space $\mathbb{R}^n$. We find that the long-time asymptotics of solutions to both problems coincide either in the case of small initial data in the weak $L^{n}$-space or for a certain class of large initial data.