  This paper studies inf-sup stable finite element discretizations of the
evolutionary Navier--Stokes equations with a grad-div type stabilization. The
analysis covers both the case in which the solution is assumed to be smooth and
consequently has to satisfy nonlocal compatibility conditions as well as the
practically relevant situation in which the nonlocal compatibility conditions
are not satisfied. The constants in the error bounds obtained do not depend on
negative powers of the viscosity. Taking into account the loss of regularity
suffered by the solution of the Navier--Stokes equations at the initial time in
the absence of nonlocal compatibility conditions of the data, error bounds of
order $\mathcal O(h^2)$ in space are proved. The analysis is optimal for
quadratic/linear inf-sup stable pairs of finite elements. We also consider the
analysis of the fully discrete case with the backward Euler method as time
integrator.
