  We prove the ill-posedness for the Leray-Hopf weak solutions of the
incompressible and ipodissipative Navier-Stokes equations, when the power of
the diffusive term $(-\Delta)^\gamma$ is $\gamma < \frac{1}{3}$. We construct
infinitely many solutions, starting from the same initial datum, which belong
to $C_{x,t}^{\frac{1}{3}-}$ and strictly dissipate their energy in small time
intervals. The proof exploits the "convex integration scheme" introduced by C.
De Lellis and L. Sz\'ekelyhidi for the incompressible Euler equations, joining
these ideas with new stability estimates for a class of non-local
advection-diffusion equations and a local (in time) well-posedness result for
the fractional Navier-Stokes system. Moreover we show the existence of
dissipative H\"older continuous solutions of Euler equations that can be
obtained as a vanishing viscosity limit of Leray-Hopf weak solutions of a
suitable fractional Navier-Stokes equations.
