An extension of an unicity class for Navier-Stokes equations

TL;DR

This paper extends the class of initial conditions under which weak solutions to the Navier-Stokes equations are known to be unique, broadening the mathematical understanding of solution behavior.

Contribution

It proves that the uniqueness criterion for weak Leray solutions applies to a larger range of regularity indices than previously established.

Findings

Uniqueness holds for $r$ in $]-1,-\frac{1}{2}]$

Extends previous results for $r$ in $]-\frac{1}{2},1]$

Broadens the class of initial data ensuring solution uniqueness.

Abstract

This is a translation from French of my paper [R. May, Extension d'une classe d'unicite pour les equations de Navier-Stokes, Ann. I. H. Poincar\'{e}-AN 27 (2010) 705-718. doi:10.1016/j.anihp.2009.11.007]. Q. Chen, C. Miao, and Z. Zhang \cite{CMZ} have proved that weak Leray solutions of the Navier-Stokes are unique in the class $L^{\frac{2}{1+r}% }([0,T].B_{\infty}^{r,\infty}(\mathbb{R}^{3})$ with $r\in]-\frac{1}{2},1].$ In this paper, we establish that this criterion remains true for $r\in ]-1,-\frac{1}{2}].$