Stability of stationary solutions to the Navier–Stokes equations in the Besov space

We consider the stability of the stationary solution w of the Navier–Stokes equations in the whole space $$\mathbb {R}^n$$ for $$n \ge 3$$. It is clarified that if w is small in $$\dot{B}^{-1+\frac{n}{p_\ast }}_{p_\ast , q^{\prime }}$$ for $$1 \le p_\ast <n$$ and $$1 < q^{\prime } \le 2$$, then for every small initial disturbance $$a \in \dot{B}^{-1+ \frac{n}{p_0}}_{p_0,q}$$ with $$1 \le p_0<n$$ and $$2\le q < \infty$$ ($$1/q + 1/q^{\prime } =1$$), there exists a unique solution $$v(t)$$ of the nonstationary Navier–Stokes equations on (0, ∞) with $$v(0) = w+a$$ such that $$\Vert v(t) - w\Vert _{L^r}=O(t^{-\frac{n}{2}(\frac{1}{n} - \frac{1}{r})})$$ and $$\Vert v(t) - w\Vert _{\dot{B}^s_{p, q}} =O(t^{-\frac{n}{2}(\frac{1}{n} - \frac{1}{p})-\frac{s}{2}})$$ as $$t\rightarrow \infty$$, for $$p_0 \le p <n$$, $$n < r < \infty$$, and small $$s > 0$$.