Existence of a Stationary Navier-Stokes Flow Past a Rigid Body, with Application to Starting Problem in Higher Dimensions

TL;DR

This paper proves the existence of stationary Navier-Stokes flows around a rigid body in higher dimensions and demonstrates convergence of nonstationary flow to these solutions, extending previous 3D results to higher dimensions with new convergence rates.

Contribution

It extends the existence and convergence results of stationary Navier-Stokes flows past a rigid body to higher dimensions, including new convergence rate estimates even in 3D.

Findings

Existence of small stationary solutions with optimal decay in higher dimensions.

Convergence of nonstationary flow to stationary solutions as time approaches infinity.

New convergence rate estimates depending on the summability of stationary solutions.

Abstract

We consider the large time behavior of the Navier-Stokes flow past a rigid body in $\mathbb{R}^n$ with $n\geq 3$. We first construct a small stationary solution possessing the optimal summability at spatial infinity, which is the same as that of the Oseen fundamental solution. When the translational velocity of the body gradually increases and is maintained after a certain finite time, we then show that the nonstationary fluid motion converges to the stationary solution corresponding to a small terminal velocity of the body as time $t\rightarrow\infty$ in $L^q$ with $q\in[n,\infty]$. This is called Finn's starting problem and the three-dimensional case was affirmatively solved by Galdi, Heywood and Shibata $(1997).$ The present paper extends their result to the case of higher dimensions. Even for the three-dimensional case, our theorem provides new convergence rate, that is determined by the summability of the stationary solution at infinity and seems to be sharp.