Uniqueness criterion of weak solutions for the 3D Navier-Stokes equations

In this paper we establish a new uniqueness result of weak solutions for the 3D Navier-Stokes equations. Under assumption that there is not uniqueness of weak solution in singular time, we prove that if two weak solutions $u$ and $v$ of 3D Navier-Stokes equations belong to $L^{\frac{5}{2}}\left( 0,T;V\right) $ with the same initial datum, then we get $u=v$. In the class $L^{3}\left( 0,T;V\right) $, we prove that $u-v\in$ $C^{0}\left( 0,T;H\right) $ and $u=v\ $when $u_{0}=v_{0}$.