Rôle de léspace de Besov ${\mathbf{B}}_{\infty }^{-1,\infty }$dans le contrôle de léxplosion èventuelle en temps fini des solutions régulières des équations de Navier-Stokes

Abstract

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

4 pages