  For initial datum of finite kinetic energy, Leray has proven in 1934 that
there exists at least one global in time finite energy weak solution of the 3D
Navier-Stokes equations. In this paper we prove that weak solutions of the 3D
Navier-Stokes equations are not unique in the class of weak solutions with
finite kinetic energy. Moreover, we prove that Holder continuous dissipative
weak solutions of the 3D Euler equations may be obtained as a strong vanishing
viscosity limit of a sequence of finite energy weak solutions of the 3D
Navier-Stokes equations.
