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 $L^\infty((0; T)\times R^n)$ and $\pi(1 + |x|)^{-n-1}\in L^1(0,T;L^1(R^n)), n\geq2.$ The interest to our theorem is twofold. From one side, the theorem is motivated by the fact that a possible pressure field $\pi$ belonging to $L^1(0,T;BMO)$ satisfies in a suitable sense our assumption on the pressure (this give an improvement to a recent previous result published on Archive for Rational Mech. Anal.). From the other side, the assumptions on the pressure appear sharps by means of a counterexample. Finally, the theorem gives the well-psedeness in a large set of solutions which are considered physically meaning.