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This paper is essentially a translation from French of my article \cite{M1}published in 2003. Let $u\in C([0,T^{\ast}[;L^{3}(\mathbb{R}% ^{3})) $ be amaximal solution of the Navier-Stokes equations. We prove that $u$ is$C^{\infty}$ on $]0,T^{\ast}[\times \mathbb{R}^{3}$ and there exists a constant$\varepsilon_{\ast}>0$ independent of $u$ such that if $T^{\ast}$ is finitethen, for all $\omega \in \overline{S(\mathbb{R%}^{3})}^{B_{\infty}^{-1,\infty}},$ we have $\overline{\lim_{t\to T^{\ast}}}\Vert u(t)-\omega\Vert_{\mathbf{B}_{\infty}^{-1,\infty}}\geq \varepsilon_{\ast}. $

The role of the Besov space $\mathbf{B}_{\infty}^{-1,\infty}$% in the  control of the eventual explosion in finite time of the regular solutions of  the Navier-Stokes equations