Abstract

In this note we give a uniqueness theorem for solutions $(u,\pi)$ to the Navier-Stokes Cauchy problem, assuming that $u$ belongs to $L^\infty((0,T)\times\Bbb R^n)$ and $(1+|x|)^{-n-1}\pi\in L^1(0,T;L^1(\R^n))$, $n\geq2$. The interest to 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 on the pressure, and by the fact that the proof is very simple.

Keywords: Uniqueness; Weak solutions; Navier-Stokes equations.

MSC: 35Q30, 76D05, 76N10

Pages:  83$-$93     

Volume  61 ,  Issue  1 ,  2009