Uniqueness of weak solutions of the Navier-Stokes equations

summary:Consider the Navier-Stokes equation with the initial data $a\in L_{\sigma }^2( \Bbb R^d) $. Let $u$ and $v$ be two weak solutions with the same initial value $a$. If $u$ satisfies the usual energy inequality and if $\nabla v\in L^2(( 0,T) ;\dot X _1(\Bbb R^d)^d)$ where $\dot X_1(\Bbb R^d)$ is the multiplier space, then we have $u=v$