  This article studies the uniqueness of the weak solution of the
incompressible Navier-Stokes Equations in the 3-dimensional case. Here, the
investigation is provided using two different approaches. The first (the main)
result is obtained for given functions possessing a certain smoothness using
the new approach. The second result is without the complementary conditions but
is, in some sense, the "local" result investigated by another approach. In
addition, here the solvability and uniqueness of the weak solutions to
auxiliary problems lead out from the main problem are investigated.
