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$L^ınfty((0,T)\times\Bbb R^n)$ and $(1+|x|)^{-n-1}\piın L^1(0,T;L^1(\R^n))$, $n\geq2$. The interestto our theorem is motivated by the fact that a possible pressure field $\widetilde \pi$, 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.