We consider the large time behavior of the Navier–Stokes flow past a rigid body in \(\mathbb {R}^n\) with \(n\ge 3\). 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 et al. (Arch Ration Mech Anal 138: 307–318, 1997). The present paper extends Galdi et al. (1997) 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.

我们考虑了Navier–Stokes流经过\（\ mathbb {R} ^ n \）中带有\（n \ ge 3 \）的刚体的长时间行为。我们首先构造一个小的固定解，它在空间无穷大处具有最佳的可加性，这与Oseen基本解相同。当身体的平移速度逐渐增加并在一定的时间后保持不变时，我们证明，随着时间\（t \ rightarrow \ infty \ ）在\（L ^ q \）中与\（q \ in [n，\ infty] \）。这被称为Finn的起始问题，而Galdi等人则肯定地解决了三维情况。（Arch Ration Mech Anal 138：307-318，1997年）。本文扩展了Galdi等人。（1997年）的情况下，以更高的尺寸。即使对于三维情况，我们的定理也提供了新的收敛速度，该收敛速度由无穷大处的平稳解的可加性确定，并且似乎很尖锐。