On the uniqueness of bounded weak solutions to the Navier-Stokes Cauchy problem

Abstract

In this note we give a uniqueness theorem for solutions $(u,\pi )$to the Navier-Stokes Cauchy problem, assuming that $u$ belongs to$L^{ı}nfty((0,T)\times {\mathbb{R}}^n)$ and $(1+|x|{)}^{-n-1}\pi ın{L}^1(0,T;L^1({\text{R}}^n))$, $n\ge 2$. The interestto our theorem is motivated by the fact that a possible pressure field $\tilde{\pi }$, belonging to $L^1(0,T;\text{rm{BMO}})$, satisfies in a suitable sense our assumption onthe pressure, and by the fact that the proof is very simple.