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

Abstract

This paper is essentially a translation from French of my article \cite{M1} published in 2003. Let $u\in C([0,T^{\ast }[;L^3(\mathbb{R}$ 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 ${\epsilon }_{\ast }>0$ independent of $u$ such that if $T^{\ast }$ is finite then, for all $ω\in \overline{S(\mathbb{R%}^{3})}^{B_{\infty }^{-1,\infty}},$ we have $\overline{{\lim }_{t\to T^{\ast }}}\Vert u(t)-\omega {\Vert }_{{\mathbf{B}}_{\infty }^{-1,\infty }}\ge {\epsilon }_{\ast }.$

6 pages