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 Rn$\mathbb {R}<^>n$ for n >= 3$n \ge 3$. It is clarified that if w is small in B?p*,q '-1+np*$\dot{B}<^>{-1+\frac{n}{p_\ast }}_{p_\ast , q<^>{\prime }}$ for 1 <= p*<n$1 \le p_\ast <n$ and 1<q '<= 2$1 < q<^>{\prime } \le 2$, then for every small initial disturbance a is an element of B?p0,q-1+np0$a \in \dot{B}<^>{-1+ \frac{n}{p_0}}_{p_0,q}$ with 1 <= p0<n$1 \le p_0<n$ and 2 <= q{\prime } =1$), there exists a unique solution v(t)$v(t)$ of the nonstationary Navier-Stokes equations on (0, infinity) with v(0)=w+a$v(0) = w+a$ such that parallel to v(t)-w parallel to Lr=O(t-n2(1n-1r))$\Vert v(t) - w\Vert _{L<^>r}=O(t<^>{-\frac{n}{2}(\frac{1}{n} - \frac{1}{r})})$ and parallel to v(t)-w parallel to B?p,qs=O(t-n2(1n-1p)-s2)$\Vert v(t) - w\Vert _{\dot{B}<^>s_{p, q}} =O(t<^>{-\frac{n}{2}(\frac{1}{n} - \frac{1}{p})-\frac{s}{2}})$ as t ->infinity$t\rightarrow \infty$, for p0 <= p<n$p_0 \le p <n$, n<r0$s > 0$.