  Let $u_0\in C_0^5 ( B_{R_0})$ be divergence-free and suppose that $u$ is a
strong solution of the three-dimensional incompressible Navier-Stokes equations
on $[0,T]$ in the whole space $\mathbb{R}^3$ such that $\| u \|_{L^\infty
((0,T);H^5 (\mathbb{R}^3 ))} + \| u \|_{L^\infty ((0,T);W^{5,\infty
}(\mathbb{R}^3 ))} \leq M <\infty$. We show that then there exists a unique
strong solution $w$ to the problem posed on $B_R$ with the homogeneous
Dirichlet boundary conditions, with the same initial data and on the same time
interval for $R\geq \max(1+R_0, C(a) C(M)^{1/a} \exp ({CM^4T/a})) )$ for any
$a\in [0,3/2)$, and we give quantitative estimates on $u-w$ and the
corresponding pressure functions.
