R\^ole de l\'espace de Besov $\mathbf{B}_{\infty}^{-1,\infty}$dans le contr\^ole de l\'explosion \`eventuelle en temps fini des solutions r\'eguli\`eres des \'equations de Navier-Stokes

TL;DR

This paper investigates the role of the Besov space _{\u221e}^{-1,\u221e} in controlling potential finite-time blow-up of regular solutions to the Navier-Stokes equations, establishing a threshold condition involving this space.

Contribution

It introduces a new criterion using the Besov space _{\u221e}^{-1,\u221e} to prevent finite-time singularities in Navier-Stokes solutions, highlighting its significance in regularity analysis.

Findings

Solutions are smooth on (0,T*) imes \u211d^n.

A lower bound _* > 0 is established for the _{}^{-1,} norm difference at blow-up time.

If T* is finite, the solution's _{}^{-1,} norm approaches a threshold _*.

Abstract

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 $\overline{\lim_{t\to T^{\ast}}}\Vert u(t)-\omega \Vert_{\mathbf{B}_{\infty}^{-1,\infty}}\geq \varepsilon_{\ast}.$