The role of the Besov space B_(infty)^(-1,infty)% in the control of the eventual explosion in finite time of the regular solutions of the Navier-Stokes equations

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 a maximal 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 finite then, 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}. $