  Let $u\in C([0,T^{\ast}[;L^{n}(\mathbb{R}% ^{n})^{n})$ be a maximal solution
of the Navier-Stokes equations. We prove that $u$ is $C^{\infty}$ on
$]0,T^{\ast}[\times \mathbb{R}^{n}$ and there exists a constant $\varepsilon
_{\ast}>0$, which depends only on $n,$ such that if $T^{\ast}$ is finite then,
for all $\omega \in S(\mathbb{R}% ^{n})^{n},$ we have $\overline{\lim_{t\to
T^{\ast}}}\Vert u(t)-\omega \Vert_{\mathbf{B}_{\infty}^{-1,\infty}}\geq
\varepsilon_{\ast}.$
